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# Where tangent line = 0 (parametric)

1. Mar 29, 2008

### motornoob101

1. The problem statement, all variables and given/known data

At which point is the tanget line to the following curve horizontal?
$$y= a sin^{3}\theta$$
$$x = acos^{3}\theta$$

2. Relevant equations

3. The attempt at a solution
$$\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$$

When $$\frac{dy}{dx} = 0$$, this means that that the tanget line is horizontal.

$$\frac{dy}{dx}=0$$ when $$\frac{dy}{d\theta} = 0$$

$$\frac{dy}{d\theta} = 3asin^{2}\thetacos\theta$$

$$0=3asin^{2}\thetacos\theta$$
$$0 = sin^{2}\thetacos\theta$$
$$0 = (1-cos^{2}\theta)(cos\theta)$$
$$cos\theta = 0$$ when $$\theta = \pi/2$$ and $$\theta = 3\pi/2$$
$$1-cos^{2}\theta = 0$$ when $$\theta = 0$$
We must also find where $$dx/d\theta =0$$ since if both $$dy/d\theta$$ and $$dx/d\theta$$ = 0 at the same $$\theta$$ then we can't use that value
$$\frac{dx}{d\theta} = -3acos^{2}\theta sin\theta$$

$$dx/d\theta = 0$$ at $$\theta = 0, \pi, \pi/2$$

So therefore we can't use $$\theta = \pi/2$$ and $$\theta = 0$$ from the $$dy/d\theta$$ expression. Thus we are left with only $$\theta = 3\pi/2$$

If we sub $$\theta = 3\pi/2$$ back into the expression for x and y, we see that this corresponds to (0, -a), which means at (0, -a) the tangent line is horizontal.

However, this answer is wrong, the curve is an astroid and the astroid have horizontal tangets at (+/- a, 0).

I don't get what I did wrong, appreciate any help. Thanks

2. Mar 29, 2008

### tiny-tim

Hi motornoob!

Why didn't you stop there?

You needed $$\frac{dy}{dx} = 0$$ , but you started examining $$\frac{dy}{d\theta}$$ instead.

Just do dy/dx = 3sin^2.cos/-3cos^2.sin = … ?

3. Mar 29, 2008

### motornoob101

Well yeah of course I could have figured out what dy/dx is which is just $$-tan\theta$$ and go from there, but I want to know why the method where I figure out where $$dy/d\theta = 0$$ doesn't work. It bugs me!

4. Mar 29, 2008

### tiny-tim

ah! …well this bit was wrong:
$$cos^{2}\theta sin\theta$$ is zero at any multiple of π/2.

5. Mar 29, 2008

### motornoob101

Ah ok, so I think here is what I did wrong.
$$\frac{dx}{d\theta} = -3acos^{2}\theta sin\theta$$

I set it to on the left side

$$0 = cos^{2}\theta sin\theta$$
$$0= (1-sin^{2}\theta)sin\theta$$

$$sin\theta = 0$$ when $$\theta = 0, \pi$$

$$1-sin^{2}\theta =0$$
$$sin^{2}\theta =1$$

I think here I made my mistake, thinking that I can just write $$sin\theta = 1$$ but it turns out $$sin\theta = +/- 1$$ so $$\theta$$ here equals to $$\pi/2, 3\pi/2$$

So then with this corrected mistake, I see that $$dy/d\theta$$ equals to 0 at the same places that $$dx/d\theta$$ goes to 0, so therefore I must use the $$dy / dx$$ expression directly rather than playing with $$dy/d\theta$$ or $$dx/d\theta$$

Would this reasoning be right? Thanks.

6. Mar 29, 2008

### tiny-tim

Absolutely!

7. Mar 29, 2008

### motornoob101

Yay, thanks for all the help!


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# How To Calculate Value Of Zero Coupon Bond

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(2 days ago) A 5 year zero coupon bond is issued with a face value of $100 and a rate of 6%. Looking at the formula,$100 would be F , 6% would be r , and t would be 5 years. After solving the equation, the original price or value would be $74.73. https://www.financeformulas.net/Zero_Coupon_Bond_Value.html Category: coupon ### Zero Coupon Bond Value Calculator » Bonds4u (2 days ago) Compute the value (price) of a zero coupon bond. The calculator, which assumes semi-annual compounding, uses the following formula to compute the value of a zero-coupon bond: Value = Face Value / (1 +Yield / 2) ** Years to Maturity * 2 https://www.bonds4u.com/zero-coupon-bond-value-calculator/ Category: coupon ### How To Calculate Value Of Zero Coupon Bond | Promo Codes (1 year ago) Must-stop destination for online shopping. Let’s join this coupon marathon to win coupons and voucher codes, saving your budget. https://couponmarathon.com/ ### Zero Coupon Bond Calculator – What is the Market Price ... (2 days ago) Zero Coupon Bond Calculator Inputs. Bond Face Value/Par Value ($) - The face or par value of the bond – essentially, the value of the bond on its maturity date. Annual Interest Rate (%) - The interest rate paid on the zero coupon bond. Years to Maturity - The numbers of years until the zero coupon bond's maturity date.; Months to Maturity - The numbers of months until bond maturity (not this ...

https://dqydj.com/zero-coupon-bond-calculator/

Category:  Promo Codes

### Zero Coupon Bond Value Calculator: Calculate Price, Yield ...

(7 days ago) Calculating Yield to Maturity on a Zero-coupon Bond. YTM = (M/P) 1/n - 1. variable definitions: YTM = yield to maturity, as a decimal (multiply it by 100 to convert it to percent) M = maturity value; P = price; n = years until maturity; Let's say a zero coupon bond is issued for $500 and will pay$1,000 at maturity in 30 years.

https://calculator.me/savings/zero-coupon-bonds.php

Category:  coupon

### Zero Coupon Bond Calculator - Miniwebtool

(2 days ago) The Zero Coupon Bond Calculator is used to calculate the zero-coupon bond value. Zero Coupon Bond Definition. A zero-coupon bond is a bond bought at a price lower than its face value, with the face value repaid at the time of maturity. It does not make periodic interest payments. When the bond reaches maturity, its investor receives its face value.

https://miniwebtool.com/zero-coupon-bond-calculator/

Category:  coupon

### How To Calculate Value Of Zero Coupon Bond | Coupon Codes

(1 year ago) Here come the coupons, coupon codes, discounts and discount codes of wide-ranging brands, the smartest way to save off. Find all in this coupon offering site.

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### How do I Calculate Zero Coupon Bond Yield? (with picture)

(5 days ago) The zero coupon bond yield is easier to calculate because there are fewer components in the present value equation. It is given by Price = (Face value)/(1 + y) n , where n is the number of periods before the bond matures.

https://www.wise-geek.com/how-do-i-calculate-zero-coupon-bond-yield.htm

Category:  Coupon Codes,  Discount Codes

### How to Calculate a Zero Coupon Bond Price | Double Entry ...

(2 days ago) The time value of money formulas can be used to calculate a zero coupon bond price. A business will issue zero coupon bonds when it wants to obtain funding from long term investors by way of debt finance.

https://www.double-entry-bookkeeping.com/bonds-payable/zero-coupon-bond-price/

Category:  coupon

### Calculating Yield to Maturity of a Zero-Coupon Bond

(2 days ago) Time value of money formulas usually require interest rate figures for each point in time. This consequently renders the yield to maturity easier to calculate for zero-coupon bonds. There are no ...

Category:  coupon

### Zero Coupon Bond Yield - Formula (with Calculator)

(2 days ago) The zero coupon bond effective yield formula is used to calculate the periodic return for a zero coupon bond, or sometimes referred to as a discount bond. A zero coupon bond is a bond that does not pay dividends (coupons) per period, but instead is sold at a discount from the face value. For example, an investor purchases one of these bonds at ...

https://www.financeformulas.net/Zero-Coupon-Bond-Effective-Yield.html

Category:  coupon

### How to Calculate PV of a Different Bond Type With Excel

(2 days ago) A. Zero Coupon Bonds . Let's say we have a zero coupon bond (a bond which does not deliver any coupon payment during the life of the bond but sells at a discount from the par value) maturing in 20 ...

https://www.investopedia.com/articles/investing/051315/present-value-different-bond-types-using-excel.asp

Category:  coupon

### Zero-Coupon Bond Value | Formula, Example, Analysis ...

(4 days ago) The zero-coupon bond value is affected by interest rate risk but not by reinvestment risk. Zero-Coupon Bond Value Calculator. You can use the zero-coupon bond value calculator below to quickly measure how much your zero-coupon bond is currently worth, by entering the required numbers.

https://studyfinance.com/zero-coupon-bond-value/

Category:  coupon

### Zero-Coupon Bond - Definition, How It Works, Formula

(2 days ago) Therefore, a zero-coupon bond must trade at a discount because the issuer must offer a return to the investor for purchasing the bond. Pricing Zero-Coupon Bonds. To calculate the price of a zero-coupon bond, use the following formula: Where: Face value is the future value (maturity value) of the bond;

Category:  coupon

### Zero Coupon Bond (Definition, Formula, Examples, Calculations)

(2 days ago) Basis Zero-Coupon Bond Regular Coupon Bearing Bond: Meaning: It refers to fixed Income security, which is sold at a discount to its Par value and doesn’t involve any cash flow during the life of the Bond except on maturity.: It refers to fixed Income security, which involves regular payment in the form of coupons and may be issued at a discount or premium depending upon market dynamism.

https://www.wallstreetmojo.com/zero-coupon-bond/

Category:  coupon

### Zero Coupon Bond Value Calculator - buyupside.com

(2 days ago) The calculator, which assumes semi-annual compounding, uses the following formula to compute the value of a zero-coupon bond: Value = Face Value / (1 +Yield / 2) ** Years to Maturity * 2 . Related Calculators. Bond Convexity Calculator. Bond Duration Calculator - Macaulay Duration, Modified Macaulay Duration and Convexity Bond Present Value ...

Category:  coupon

### Zero-Coupon Bond - Investopedia

(2 days ago) Zero-Coupon Bond: A zero-coupon bond is a debt security that doesn't pay interest (a coupon) but is traded at a deep discount, rendering profit at maturity when the bond is redeemed for its full ...

https://www.investopedia.com/terms/z/zero-couponbond.asp

Category:  coupon

(2 days ago) Divide the face value of the bond to calculate the price to pay for the zero-coupon bond to achieve your desired rate of return. Zero-Coupon Bond Price Example For example, say you want to earn a 6 percent rate of return per year on a bond with a face value of $2,000 that will mature in two years. https://budgeting.thenest.com/calculate-price-zero-coupon-bond-4203.html Category: coupon ### Zero Coupon Bond: Definition, Formula & Example - Video ... (3 days ago) The basic method for calculating a zero coupon bond's price is a simplification of the present value (PV) formula. The formula is price = M / (1 + i )^ n where: M = maturity value or face value https://study.com/academy/lesson/zero-coupon-bond-definition-formula-example.html Category: coupon ### Zero Coupon Bond Value Calculator | Calculate Zero Coupon ... (4 days ago) Zero Coupon Bond Value is referred to as a pure discount bond or simply discount bond, is a bond that does not pay coupon payments, and instead pays one lump sum at maturity is calculated using Zero Coupon Bond Value=Face Value/(1+Rate of Return)^Time to Maturity.To calculate Zero Coupon Bond Value, you need Face Value (F), Rate of Return (RoR) and Time to Maturity (T). https://www.calculatoratoz.com/en/zero-coupon-bond-value-calculator/Calc-240 Category: coupon ### Zero Coupon Bond Calculator - Zero Coupon Bond Formula (3 days ago) The following zero coupon bond formula shows how to calculate zero coupon bond yield. Zero Coupon Bond Value = F / (1 + r/100)^n, where F = Bond Face Value, r = Rate, n = Years to Maturity. Bond Equivalent Yield Calculator Yield to Maturity Calculator Bond Yield Calculator Effective Yield Calculator Dividend Yield Calculator: Electrical Calculators https://online-calculator.org/zero-coupon-bond-calculator.aspx Category: coupon ### Zero Coupon Bond Sale Calculations, Accounting & Journal ... (13 days ago) How to calculate and record the sale of a zero coupon bond, start with a cash flow diagram, face (maturity) value, no stated rate of interest on bond and no ... https://www.youtube.com/watch?v=y_pVfjAZc4E Category: coupon ### How to calculate bond price in Excel? (3 days ago) Calculate price of a zero coupon bond in Excel For example there is 10-years bond, its face value is$1000, and the interest rate is 5.00%. Before the maturity date, the bondholder cannot get any coupon as below screenshot shown.

https://www.extendoffice.com/documents/excel/5088-excel-calculate-bond-price.html

Category:  coupon

(2 days ago) Since the coupon rate is higher than the YTM, the bond price is higher than the face value, and as such, the bond is said to be traded at a premium. Example #3. Let us take the example of a zero-coupon bond. Let us assume a company QPR Ltd has issued a zero-coupon bond with having a face value of $100,000 and maturing in 4 years. https://www.wallstreetmojo.com/bond-pricing-formula/ Category: coupon ### wikiHow: How-to instructions you can trust. (2 days ago) wikiHow: How-to instructions you can trust. https://www.wikihow.com/Calculate-a-zero-coupon-bond Category: coupon ### Accounting for Zero-Coupon Bonds (3 days ago) A zero-coupon bond is a bond which does not pay any periodic interest but whose total return results from the difference between its issuance price and maturity value. For example, if Company Z issues 1 million bonds of$1000 face value bonds due to maturity in 5 years but which do not pay any interest, it is a zero-coupon bond.

https://xplaind.com/379952/zero-coupon-bond

Category:  coupon

### Compound Accreted Value (CAV)

(2 days ago) Compound Accreted Value - CAV: A measure of the theoretical value of a zero-coupon bond at any given point in time. Because there are no interest payments like there are with traditional bonds ...

https://www.investopedia.com/terms/c/cav.asp

Category:  coupon

(3 days ago) The company plans to issue 5,000 such bonds, and each bond has a par value of $1,000 with a coupon rate of 7%, and it is to mature in 15 years. The effective yield to maturity is 9%. Determine the price of each bond and the money to be raised by XYZ Ltd through this bond issue. Below is given data for the calculation of the coupon bond of XYZ Ltd. https://www.wallstreetmojo.com/coupon-bond-formula/ Category: coupon ### Bond Yield to Maturity (YTM) Calculator - DQYDJ (2 days ago) A zero coupon bond is a bond which doesn't pay periodic payments, instead having only a face value (value at maturity) and a present value (current value). This makes calculating the yield to maturity of a zero coupon bond straight-forward: Let's take the following bond as an example: Current Price:$600;

https://dqydj.com/bond-yield-to-maturity-calculator/

Category:  coupon

### How do you calculate the rate on a zero coupon bond ...

(1 days ago) The pice of zero coupon bond will be determine by following formula: {eq}Price=\displaystyle \frac{Par\:value}{(1+I)^N} {/eq} Hence, the yield will be estimated by following formula:

Category:  coupon

### Coupon Rate Formula | Step by Step Calculation (with Examples)

(1 days ago) Formula to Calculate Coupon Rate. Coupon Rate Formula is used for the purpose of calculating the coupon rate of the bond and according to the formula coupon rate of the bond will be calculated by dividing the total amount of annual coupon payments with the par value of the bonds and multiplying the resultant with the 100.

https://www.wallstreetmojo.com/coupon-rate-formula/

Category:  coupon

### Calculate the Value of Your Paper Savings Bond(s)

(2 days ago) The Savings Bond Calculator WILL: Calculate the value of a paper bond based on the series, denomination, and issue date entered. (To calculate a value, you don't need to enter a serial number. However, if you plan to save an inventory of bonds, you may want to enter serial numbers.)

https://www.treasurydirect.gov/BC/SBCPrice

Category:  coupon

### Zero Coupon Bond Yield Calculator - YTM of a discount bond

(1 days ago) Zero Coupon Bond Yield Calculator. A Zero Coupon Bond or a Deep Discount Bond is a bond that does not pay periodic coupon or interest. These bonds are issued at a discount to their face value and therefore the difference between the face value of the bond and its issue price represents the interest yield of the bond.

https://vindeep.com/Corporate/ZeroCouponBondYieldCalc.aspx

Category:  coupon

### Calculation of the Value of Bonds (With Formula)

(2 days ago) (c) An investor buys 100, 13% debentures and for 100 zero bonds. (d) Assume that the investor sells the shares at Rs. 75/-. He bought the zero bonds at Rs. 10 face value and Rs. 35/- premium, i.e., for Rs. 45 each. This example shows that after the taxes have been paid, the investor has a higher return in zero bonds rather than in debentures.

https://www.yourarticlelibrary.com/accounting/bonds/calculation-of-the-value-of-bonds-with-formula/82406

Category:  coupon

### Bond Valuation Definition - Investopedia

(2 days ago) Bond valuation is a technique for determining the theoretical fair value of a particular bond. Bond valuation includes calculating the present value of the bond's future interest payments, also ...

https://www.investopedia.com/terms/b/bond-valuation.asp

Category:  coupon

### Zero Coupon Bond Calculator - miniwebtool.com

(3 months ago) The Zero Coupon Bond Calculator is used to calculate the zero-coupon bond value. Zero Coupon Bond Definition. A zero-coupon bond is a bond bought at a price lower than its face value, with the face value repaid at the time of maturity. It does not make periodic interest payments. When the bond reaches maturity, its investor receives its face value.

https://miniwebtool.com/zero-coupon-bond-calculator/?f=1000&r=9&t=20

Category:  coupon

### Accounting for Zero-Coupon Bonds - GitHub Pages

(3 days ago) Explain how interest is earned on a zero-coupon bond. Understand the method of arriving at an effective interest rate for a bond. Calculate the price of a zero-coupon bond and list the variables that affect this computation. Prepare journal entries for a zero-coupon bond using the effective rate method. Explain the term “compounding.”

https://saylordotorg.github.io/text_financial-accounting/s17-03-accounting-for-zero-coupon-bon.html

Category:  coupon

### Zero Coupon Bond Effective Yield Calculator | Calculate ...

(7 days ago) Zero Coupon Bond Effective Yield is used to calculate the periodic return for a zero coupon bond, or sometimes referred to as a discount bond and is represented as ZCB Yield=(F/PV)^(1/n)-1 or Zero Coupon Bond Effective Yield=(Face Value/Present Value)^(1/Number of Periods)-1.Face value is the nominal value or dollar value of a security stated by the issuer, The number of periods is the periods ...

https://www.calculatoratoz.com/en/zero-coupon-bond-effective-yield-calculator/Calc-241

Category:  coupon

### How to Calculate a Coupon Payment: 7 Steps (with Pictures)

(5 days ago) If you know the face value of the bond and its coupon rate, you can calculate the annual coupon payment by multiplying the coupon rate times the bond's face value. For example, if the coupon rate is 8% and the bond's face value is $1,000, then the annual coupon payment is .08 * 1000 or$80.

https://www.wikihow.com/Calculate-a-Coupon-Payment

Category:  coupon

### Zero-Coupon Bond Value Calculator - MYMATHTABLES.COM

(2 days ago) A zero-coupon bond is a debt security that does not pay interest but instead trades at a deep discount, rendering a profit at maturity, when the bond is redeemed for its full face value. P = m (1 + r) n. Where, P = Zero-Coupon Bond Price. M = Face value at maturity or face value of bond. r = annual yield or rate . n = years until maturity ...

https://www.mymathtables.com/calculator/finance/zero-coupon-bond-price-calculator.html

Category:  coupon

### How Do I Calculate Gains from a Zero-Coupon Bond Fund ...

(2 days ago) The formula for calculating how much you'll wind up with when your zero-coupon bond fund matures (akin to the face value of a bond) if you invest a certain amount now (akin to the price of a bond ...

https://www.thestreet.com/investing/funds/how-do-i-calculate-gains-from-a-zero-coupon-bond-fund-718794

Category:  coupon

### How to Calculate Bond Value: 6 Steps (with Pictures) - wikiHow

(2 years ago) A bond is a debt security that pays a fixed amount of interest until maturity. When a bond matures, the principal amount of the bond is returned to the bondholder. Many investors calculate the present value of a bond. The present value (i.e. the discounted value of a future income stream) is used ...

https://www.wikihow.com/Calculate-Bond-Value

Category:  coupon

(2 days ago) The Par Value of each Bond is $100. Based on the details furnished below to let’s try to calculate the value of the Portfolio’s DV01 and understand the resultant impact: The calculation is as follows: Dollar Value of One Basis Point = Dollar Duration *$1000000*0.0001 = $85.84*$1000000*0.0001 = $8,584 https://www.wallstreetmojo.com/dv01/ Category: coupon ### Zero Coupon Bond Calculate Value - uberbestorder.info (3 days ago) This model is replacing the 15 Zero Coupon Bond Calculate Value in model. This has the same length and width, but bit thicker and about 0.3 lb heavier. You are just getting bigger display on same footprint, and mush Zero Coupon Bond Calculate Value better keyboard as compare to latest 15 Zero Coupon Bond Calculate Value in model. https://uberbestorder.info/zero-coupon-bond-calculate-value.htm Category: coupon ### Zero Coupon Bond Calculator Excel (4 months ago) How to calculate bond price in Excel? CODES (3 days ago) Calculate price of a zero coupon bond in Excel For example there is 10-years bond, its face value is$1000, and the interest rate is 5.00%. Before the maturity date, the bondholder cannot get any coupon as below screenshot shown.

https://www.mybestcouponcodes.com/zero-coupon-bond-calculator-excel/

Category:  coupon

### What Is Coupon Rate and How Do You Calculate It?

(4 days ago) Instead, they sell the bond below par value. At maturity, the bond holder redeems the bond for its entire par value. The note’s rate of return is the difference between its sale price and its price at maturity. For example, ABC Corporation could issue a 10-year, zero-coupon bond with a par value of $1,000. They might then sell it for$900.

https://smartasset.com/investing/bond-coupon-rate

Category:  coupon

### Coupon Bond Formula | Examples with Excel Template

(4 days ago) Coupon Bond = $25 * [1 – (1 + 4.5%/2)-16] + [$1000 / (1 + 4.5%/2) 16; Coupon Bond = $1,033; Therefore, the current market price of each coupon bond is$1,033, which means it is currently traded at a premium (current market price higher than par value). Explanation. The formula for coupon bond can be derived by using the following steps:

https://www.educba.com/coupon-bond-formula/

Category:  coupon

### How to Calculate Bond Interest Payments | Budgeting Money ...

(1 days ago) If you know the current value of a zero-coupon bond, you can skip calculating the interest payments that have accrued since the bond was issued. Current value is the value of the bond at the end of the most recent compounding period. Just multiply that current value by the periodic interest rate to calculate the interest payment.

https://budgeting.thenest.com/calculate-bond-interest-payments-3347.html

Category:  coupon

### Zero Coupon Bond Price Calculator - allcoupons.org

(4 days ago) Zero Coupon Bond Calculator - Miniwebtool. CODES (3 days ago) The Zero Coupon Bond Calculator is used to calculate the zero-coupon bond value. Zero Coupon Bond Definition. A zero-coupon bond is a bond bought at a price lower than its face value, with the face value repaid at the time of maturity.

https://allcoupons.org/zero-coupon-bond-price-calculator/

Category:  coupon

### How to Calculate Bond Discount Rate: 14 Steps (with Pictures)

(2 days ago) Bonds are sold at a discount when the market interest rate exceeds the coupon rate of the bond. In order to calculate how the amount of the bond discount, you need to need to calculate the present value of the principal and the present value of the coupon payments. ... and the face value of the bond. First, calculate the bond’s market price ...

https://www.wikihow.com/Calculate-Bond-Discount-Rate

Category:  coupon

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## 17Calculus Infinite Series - Limit Comparison Test

The Limit Comparison Test (LCT) and the Direct Comparison Test are two tests where you choose a series that you know about and compare it to the series you are working with to determine convergence or divergence. These two tests are the next most important, after the Ratio Test, and it will help you to know these well. They are very powerful and fairly easy to use.

If you would like a complete lecture on the Limit Comparison Test, we recommend this video clip. As the title implies, this video starts with the (Direct) Comparison Test but we have the video start when he begins discussing the Limit Comparison Test.

### Prof Leonard - Calculus 2 Lecture 9.4: The Comparison Test for Series and The Limit Comparison Test

video by Prof Leonard

Limit Comparison Test

For the two series $$\sum{a_n}$$ and $$\sum{t_n}$$ where $$a_n > 0$$ and $$t_n > 0$$, we calculate $$\displaystyle{\lim_{n \to \infty}{\left[ \frac{a_n}{t_n} \right]} = L}$$
1. when L is finite and positive, the two series either both converge or both diverge
2. when $$L=0$$ and $$\sum{t_n}$$ converges, then $$\sum{a_n}$$ also converges
3. when $$L=\infty$$ and $$\sum{t_n}$$ diverges, then $$\sum{a_n}$$ also diverges
4. if the limit does not exist then the test is inconclusive

What The Limit Comparison Test Says

This test is pretty straightforward. In our notation, we say that the series that you are trying to determine whether it converges or diverges is $$\sum{a_n}$$ and the test series that you know whether it converges or diverges is $$\sum{t_n}$$. The limit $$L$$ has be defined for you come to any conclusion.

Also, notice that the fraction can be inverted and the test still works for case 1 (but not cases 2 and 3). For example, if you get $$L=3$$ for one fraction, then you would get $$L=1/3$$ for the other fraction. Both are finite and positive and both will tell you whether your series converges or diverges. If you invert the fraction then cases 2 and 3 will change. So it is important to check your fraction if you are trying to apply cases 2 or 3.

When To Use The Limit Comparison Test

This test can't be used all the time. Here is what to check before trying this test.
- As with all theorems, the conditions for the test must be met. The main one is that $$a_n > 0$$. If this isn't the case with your series, don't stop yet. Look at the Absolute Convergence Theorem to see if that will help you.
- A second important thing to consider is whether or not the limit can be evaluated. For example, if you have a term that oscillates, like sine or cosine, then you can't evaluate the limit. So this test won't help you. In this case, the Direct Comparison Test might work better.

How To Choose A Test Series

When you are first learning this technique, it may look like the test series comes out of thin air and you just randomly choose one and see if it works. If it doesn't, you try another one. This is not the best way to choose a test series. The best way I've found is to use the series you are asked to work with and come up with the test series. There are several things to consider.

The first key is to choose a test series that you know converges or diverges AND that will help you get a finite, positive limit.
Idea 1: If you have polynomials in both the numerator and denominator of a fraction, drop all terms except for the highest power terms (in both parts) and simplify. Drop any constants. What you end up with may be a good comparison series. The reason this works is that, as n gets larger and larger, the highest powers dominate. You will often end up with a p-series that you know either converges or diverges.
Idea 2: Choose a p-series or geometric series since you can tell right away whether it converges or diverges.
Idea 3: If you have a sine or cosine term, you are always guaranteed that the result is less than or equal to one and greater than or equal to negative one. If you don't have any bounds on the angle, these are the best you can do. So replace the sine or cosine term with one.
Idea 4: If you have a natural log, use the fact that $$\ln(n) \leq n$$ for $$n\geq1$$ to replace $$\ln(n)$$ with n or use $$\ln(n) \geq 1$$ for $$n\geq3$$ .

As you get experience with this test, it will become easier to determine a good test series. So work plenty of practice problems and follow the advice in the infinite series study techniques section.

Before getting started working practice problems, watching these two video clips will give you a better feel for the limit comparison test. Both clips are short and to the point.

### Dr Chris Tisdell - Series, comparison + ratio tests [2min-19secs]

video by Dr Chris Tisdell

### PatrickJMT - Direct Comparison Test / Limit Comparison Test for Series - Basic Info [1min-35secs]

video by PatrickJMT

Limit Comparison Test Proof

Here is a video showing a proof of the Limit Comparison Test. You do not need to watch this in order to understand and use the Limit Comparison Test. However, we include it here for those of you who are interested.

### Linda Green - Proof of the Limit Comparison Test

video by Linda Green

Practice

Unless otherwise instructed, determine the convergence or divergence of the following series using the limit comparison test (LCT), if possible. [Of course, if the LCT is inconclusive, use another test to determine convergence or divergence.]

Basic

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n}{n^2+1} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n}{n^2+1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series diverges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n}{n^2+1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

There are several ways to determine convergence or divergence of this infinite series. First, we notice that as n gets very, very large, the 1 in the denominator becomes negligible when compared to $$n^2$$. So the fraction is getting closer and closer to $$n/n^2 = 1/n$$. So let's try comparing the original series with $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n} } }$$.
I find the Limit Comparison Test to be the easiest to use. So let's set up the limit.

 $$\displaystyle{ \lim_{n \to \infty}{ \left( \frac{n}{n^2+1} \div \frac{1}{n} \right) } }$$ $$\displaystyle{ \lim_{n \to \infty}{ \left( \frac{n}{n^2+1} \cdot \frac{n}{1} \right) } }$$ $$\displaystyle{ \lim_{n \to \infty}{\frac{n^2}{n^2+1}} = 1 }$$

Since the limit is positive and finite, the two series either both converge or both diverge. The series $$\sum{1/n}$$ diverges since it is a p-series with $$p=1$$, which means the original series also diverges by the Limit Comparison Test.
Note - The Direct Comparison Test and Integral Test also work.

### 133 video

video by Ludus

The series diverges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=0}^{\infty}{ \frac{3}{n^2+1} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=0}^{\infty}{ \frac{3}{n^2+1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series converges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=0}^{\infty}{ \frac{3}{n^2+1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

There are several ways to determine convergence or divergence of this infinite series. First, we notice that as n gets very, very large, the 1 in the denominator becomes negligible when compared to $$n^2$$. So the fraction is getting closer and closer to $$1/n^2$$. So let's try comparing the original series with $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }$$.
I find the Limit Comparison Test to be the easiest to use. So let's set up the limit.

 $$\displaystyle{ \lim_{n \to \infty}{ \left( \frac{3}{n^2+1} \div \frac{1}{n^2} \right) } }$$ $$\displaystyle{ \lim_{n \to \infty}{ \left( \frac{3}{n^2+1} \cdot \frac{n^2}{1} \right) } }$$ $$\displaystyle{ \lim_{n \to \infty}{\frac{3n^2}{n^2+1}} = 3 }$$

Since the limit is positive and finite, the two series either both converge or both diverge. The series $$\sum{1/n^2}$$ converges since it is a p-series with $$p=2$$, which means the original series also converges by the Limit Comparison Test.
The Direct Comparison Test and Integral Test may also work.

The series converges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=0}^{\infty}{ \frac{5n+2}{n^3+1} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=0}^{\infty}{ \frac{5n+2}{n^3+1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series converges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=0}^{\infty}{ \frac{5n+2}{n^3+1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 135 video

video by MIT OCW

The series converges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n+3}{n(n+2)} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n+3}{n(n+2)} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 3214 video

video by Ludus

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n+1}{2n^3-n^2} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n+1}{2n^3-n^2} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

We skip over the first minute and a half or so of this video to where he actually starts solving this problem. However, in that first minute and a half, he discusses why the Direct Comparison Test does not work to solve this problem. If you have had the Direct Comparison Test, you may want to go back and watch that first part of the video. However, you do not need that first minute and a half in order to solve this problem with the LCT.

### 3203 video

video by Trefor Bazett

$$\displaystyle{ \sum_{n=0}^{\infty}{ \frac{1}{\sqrt{n^2+2}} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=0}^{\infty}{ \frac{1}{\sqrt{n^2+2}} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 3201 video

$$\displaystyle{ \sum_{n=3}^{\infty}{ \frac{1}{\sqrt{n^2-4}} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=3}^{\infty}{ \frac{1}{\sqrt{n^2-4}} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series diverges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=3}^{\infty}{ \frac{1}{\sqrt{n^2-4}} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 136 video

video by Dr Chris Tisdell

The series diverges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2}{n^5+8} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2}{n^5+8} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 3200 video

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3n+1}{(2n-1)^4} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3n+1}{(2n-1)^4} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series converges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3n+1}{(2n-1)^4} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 138 video

video by Dr Chris Tisdell

The series converges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{6n-1}} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{6n-1}} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series diverges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{6n-1}} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 139 video

video by Dr Chris Tisdell

The series diverges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{10n^2}{n^3-1} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{10n^2}{n^3-1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series diverges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{10n^2}{n^3-1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 140 video

video by Krista King Math

The series diverges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{3^n+5} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{3^n+5} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 3202 video

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{2^n-1} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{2^n-1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 3204 video

video by Trefor Bazett

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{2^n - 7} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{2^n - 7} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 3215 video

video by Ludus

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3}{4^n-3} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3}{4^n-3} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series converges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3}{4^n-3} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 141 video

video by MIP4U

The series converges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{5}{\sqrt{n^2-1}} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{5}{\sqrt{n^2-1}} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series diverges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{5}{\sqrt{n^2-1}} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 142 video

video by MIP4U

The series diverges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2n^2-1}{3n^5+2n+1} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2n^2-1}{3n^5+2n+1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 3213 video

video by Ludus

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^4+2n^2-1}{6n^6+4n^4} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^4+2n^2-1}{6n^6+4n^4} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series converges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^4+2n^2-1}{6n^6+4n^4} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 143 video

video by MIP4U

The series converges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{2/3}+1} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{2/3}+1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series diverges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{2/3}+1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 144 video

video by Dr Phil Clark

The series diverges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2}{n^3-4} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2}{n^3-4} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series converges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2}{n^3-4} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 146 video

video by PatrickJMT

The series converges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{\sqrt{n}}{n-1} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{\sqrt{n}}{n-1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series diverges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{\sqrt{n}}{n-1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 147 video

video by PatrickJMT

The series diverges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{5}{2n^2+3} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{5}{2n^2+3} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series converges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{5}{2n^2+3} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 149 video

video by PatrickJMT

The series converges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3+1}{n^4-n+5} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3+1}{n^4-n+5} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series diverges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3+1}{n^4-n+5} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 150 video

video by PatrickJMT

The series diverges by the Limit Comparison Test.

$$\displaystyle{\sum_{n=1}^{\infty}{\frac{n+1}{3n^3(4n-1)}}}$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{\sum_{n=1}^{\infty}{\frac{n+1}{3n^3(4n-1)}}}$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series converges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{\sum_{n=1}^{\infty}{\frac{n+1}{3n^3(4n-1)}}}$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 152 video

video by PatrickJMT

The series converges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3-n+14}{n^5+n^3+1} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3-n+14}{n^5+n^3+1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series converges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3-n+14}{n^5+n^3+1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 154 video

video by PatrickJMT

The series converges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2+4} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2+4} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series converges by the limit comparison test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2+4} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

Compare to $$\sum{1/n^2}$$, which is a convergent p-series with $$p = 2$$
$$\displaystyle{a_n=\frac{1}{n^2+4}}$$, $$\displaystyle{t_n=\frac{1}{n^2}}$$

$$\begin{array}{rcl} \displaystyle{ \lim_{n\to\infty}{ \frac{a_n}{t_n} } } & = & \displaystyle{ \lim_{n\to\infty}{ \left[ \frac{1}{n^2+4} \cdot \frac{n^2}{1} \right] } } \\ & = & \displaystyle{ \lim_{n\to\infty}{ \left[ \frac{n^2}{n^2+4} \frac{1/n^2}{1/n^2} \right] } } \\ & = & \displaystyle{ \lim_{n\to\infty}{ \frac{1}{1+4/n^2} } = 1 } \end{array}$$

Since $$\displaystyle{ 0 \leq \lim_{n\to\infty}{ \frac{a_n}{t_n} } = 1 \lt \infty }$$ and $$\sum{t_n}$$ converges, the series also converges.
The limit comparison test is one of the best tests for this series. For a discussion on using other tests, see the practice problem page dedicated to this problem.

The series converges by the limit comparison test.

$$\displaystyle{ \sum_{n=1}^{\infty}{\frac{n^2-1}{n^3+4}}}$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{\frac{n^2-1}{n^3+4}}}$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series diverges by the limit comparison test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{\frac{n^2-1}{n^3+4}}}$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

Use the test series $$\sum{1/n}$$, a divergent p-series.
$$\begin{array}{rcl} \displaystyle{ \lim_{n\to\infty}{ \left| \frac{a_n}{t_n} \right| } } & = & \displaystyle{ \lim_{n\to\infty}{ \left[ \frac{n^2-1}{n^3+4} \frac{n}{1} \right] } } \\ & = & \displaystyle{ \lim_{n\to\infty}{ \left[ \frac{n^3-n}{n^3+4} \frac{1/n^3}{1/n^3} \right] } } \\ & = & \displaystyle{ \lim_{n\to\infty}{ \left[ \frac{1-1/n^2}{1+4/n^3} \right] } = 1} \\ \end{array}$$
Since the limit is finite and positive, both series either converge or diverge. Since the test series diverges, so does the original series.

The series diverges by the limit comparison test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\sqrt{n+1}}{2n^2+n+1} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\sqrt{n+1}}{2n^2+n+1} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 3208 video

video by blackpenredpen

Intermediate

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n+1}{n\sqrt{n}} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n+1}{n\sqrt{n}} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 3206 video

video by blackpenredpen

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1+3^n}{4+2^n} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1+3^n}{4+2^n} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 3217 video

video by Nakia Rimmer

$$\displaystyle{\sum_{n=1}^{\infty}{\frac{2\sqrt{n+3}}{n+5}}}$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{\sum_{n=1}^{\infty}{\frac{2\sqrt{n+3}}{n+5}}}$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series diverges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{\sum_{n=1}^{\infty}{\frac{2\sqrt{n+3}}{n+5}}}$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 151 video

video by PatrickJMT

The series diverges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{9^n}{3+10^n} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{9^n}{3+10^n} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 3207 video

video by blackpenredpen

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{e^{1/n}}{n} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{e^{1/n}}{n} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 3209 video

video by blackpenredpen

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{4n^2}{\sqrt{9n^4+1}} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{4n^2}{\sqrt{9n^4+1}} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series diverges by the Divergence Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{4n^2}{\sqrt{9n^4+1}} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 153 video

video by PatrickJMT

The series diverges by the Divergence Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2n^3+7}{n^4\sin^2(n)} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2n^3+7}{n^4\sin^2(n)} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

He chose to compare this series with $$\sum{2/n}$$. You could have used $$\sum{1/n}$$ or any constant multiple of $$\sum{1/n}$$. We probably would have chosen $$\sum{1/n}$$.

### 3216 video

video by Ludus

$$\displaystyle{ \sum_{n=1}^{\infty}{(e^{1/n}-1)} }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{(e^{1/n}-1)} }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 3218 video

video by turksvids

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\ln(n)}{n^{3/2}} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\ln(n)}{n^{3/2}} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series converges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\ln(n)}{n^{3/2}} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 137 video

video by Dr Chris Tisdell

The series converges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\sqrt[3]{n}+\ln(n^4)}{n} } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\sqrt[3]{n}+\ln(n^4)}{n} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series diverges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\sqrt[3]{n}+\ln(n^4)}{n} } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 155 video

video by PatrickJMT

The series diverges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \sin(1/n) } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \sin(1/n) } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

The series diverges by the Limit Comparison Test.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \sin(1/n) } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

This problem is solved by two different instructors.

### 156 video

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### 156 video

video by PatrickJMT

The series diverges by the Limit Comparison Test.

$$\displaystyle{ \sum_{n=1}^{\infty}{ \sin \left[ \frac{1}{n^2} \right] } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \sin \left[ \frac{1}{n^2} \right] } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Solution

### 3205 video

video by blackpenredpen

$$\displaystyle{ \sum_{n=1}^{\infty}{ ( \sqrt[n]{2} - 1 ) } }$$

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ ( \sqrt[n]{2} - 1 ) } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Hint

Choose $$\sum{1/n}$$ as your test series.

Problem Statement

Determine the convergence or divergence of the series $$\displaystyle{ \sum_{n=1}^{\infty}{ ( \sqrt[n]{2} - 1 ) } }$$. If possible, use the Limit Comparison Test. If the LCT is inconclusive, use another test to determine convergence or divergence.

Hint

Choose $$\sum{1/n}$$ as your test series.

Solution

### 3211 video

video by blackpenredpen

You CAN Ace Calculus

 infinite limits infinite series basics p-series geometric series Note: Although it is not absolutely necessary to know the Direct Comparison Test for this page, the two comparison tests are closely related. Many times (but not always), both techniques will work when determining the convergence or divergence of a series. So which technique you choose is often personal preference (assuming you are given the opportunity to choose).

### Limit Comparison Test Quick Breakdown

 used to prove convergence yes used to prove divergence yes can be inconclusive yes $$\sum{a_n}$$ is the series we are trying to determine convergence or divergence $$\sum{t_n}$$ is the test series

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text:
Outlook: Global Water Resources, Inc. is assigned short-term Ba1 & long-term Ba1 estimated rating.
Time series to forecast n: 25 Feb 2023 for (n+8 weeks)
Methodology : Statistical Inference (ML)

Abstract

Global Water Resources, Inc. prediction model is evaluated with Statistical Inference (ML) and Multiple Regression1,2,3,4 and it is concluded that the GWR:TSX stock is predictable in the short/long term. According to price forecasts for (n+8 weeks) period, the dominant strategy among neural network is: Buy

Key Points

1. Is now good time to invest?
2. Technical Analysis with Algorithmic Trading
3. What are main components of Markov decision process?

GWR:TSX Target Price Prediction Modeling Methodology

We consider Global Water Resources, Inc. Decision Process with Statistical Inference (ML) where A is the set of discrete actions of GWR:TSX stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4

F(Multiple Regression)5,6,7= $\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \dots & {p}_{1n}\\ & ⋮\\ {p}_{j1}& {p}_{j2}& \dots & {p}_{jn}\\ & ⋮\\ {p}_{k1}& {p}_{k2}& \dots & {p}_{kn}\\ & ⋮\\ {p}_{n1}& {p}_{n2}& \dots & {p}_{nn}\end{array}$ X R(Statistical Inference (ML)) X S(n):→ (n+8 weeks) $\stackrel{\to }{S}=\left({s}_{1},{s}_{2},{s}_{3}\right)$

n:Time series to forecast

p:Price signals of GWR:TSX stock

j:Nash equilibria (Neural Network)

k:Dominated move

a:Best response for target price

For further technical information as per how our model work we invite you to visit the article below:

How do AC Investment Research machine learning (predictive) algorithms actually work?

GWR:TSX Stock Forecast (Buy or Sell) for (n+8 weeks)

Sample Set: Neural Network
Stock/Index: GWR:TSX Global Water Resources, Inc.
Time series to forecast n: 25 Feb 2023 for (n+8 weeks)

According to price forecasts for (n+8 weeks) period, the dominant strategy among neural network is: Buy

X axis: *Likelihood% (The higher the percentage value, the more likely the event will occur.)

Y axis: *Potential Impact% (The higher the percentage value, the more likely the price will deviate.)

Z axis (Grey to Black): *Technical Analysis%

IFRS Reconciliation Adjustments for Global Water Resources, Inc.

1. Despite the requirement in paragraph 7.2.1, an entity that adopts the classification and measurement requirements of this Standard (which include the requirements related to amortised cost measurement for financial assets and impairment in Sections 5.4 and 5.5) shall provide the disclosures set out in paragraphs 42L–42O of IFRS 7 but need not restate prior periods. The entity may restate prior periods if, and only if, it is possible without the use of hindsight. If an entity does not restate prior periods, the entity shall recognise any difference between the previous carrying amount and the carrying amount at the beginning of the annual reporting period that includes the date of initial application in the opening retained earnings (or other component of equity, as appropriate) of the annual reporting period that includes the date of initial application. However, if an entity restates prior periods, the restated financial statements must reflect all of the requirements in this Standard. If an entity's chosen approach to applying IFRS 9 results in more than one date of initial application for different requirements, this paragraph applies at each date of initial application (see paragraph 7.2.2). This would be the case, for example, if an entity elects to early apply only the requirements for the presentation of gains and losses on financial liabilities designated as at fair value through profit or loss in accordance with paragraph 7.1.2 before applying the other requirements in this Standard.
2. Historical information is an important anchor or base from which to measure expected credit losses. However, an entity shall adjust historical data, such as credit loss experience, on the basis of current observable data to reflect the effects of the current conditions and its forecasts of future conditions that did not affect the period on which the historical data is based, and to remove the effects of the conditions in the historical period that are not relevant to the future contractual cash flows. In some cases, the best reasonable and supportable information could be the unadjusted historical information, depending on the nature of the historical information and when it was calculated, compared to circumstances at the reporting date and the characteristics of the financial instrument being considered. Estimates of changes in expected credit losses should reflect, and be directionally consistent with, changes in related observable data from period to period
3. If, in applying paragraph 7.2.44, an entity reinstates a discontinued hedging relationship, the entity shall read references in paragraphs 6.9.11 and 6.9.12 to the date the alternative benchmark rate is designated as a noncontractually specified risk component for the first time as referring to the date of initial application of these amendments (ie the 24-month period for that alternative benchmark rate designated as a non-contractually specified risk component begins from the date of initial application of these amendments).
4. For the purposes of the transition provisions in paragraphs 7.2.1, 7.2.3–7.2.28 and 7.3.2, the date of initial application is the date when an entity first applies those requirements of this Standard and must be the beginning of a reporting period after the issue of this Standard. Depending on the entity's chosen approach to applying IFRS 9, the transition can involve one or more than one date of initial application for different requirements.

*International Financial Reporting Standards (IFRS) adjustment process involves reviewing the company's financial statements and identifying any differences between the company's current accounting practices and the requirements of the IFRS. If there are any such differences, neural network makes adjustments to financial statements to bring them into compliance with the IFRS.

Conclusions

Global Water Resources, Inc. is assigned short-term Ba1 & long-term Ba1 estimated rating. Global Water Resources, Inc. prediction model is evaluated with Statistical Inference (ML) and Multiple Regression1,2,3,4 and it is concluded that the GWR:TSX stock is predictable in the short/long term. According to price forecasts for (n+8 weeks) period, the dominant strategy among neural network is: Buy

GWR:TSX Global Water Resources, Inc. Financial Analysis*

Rating Short-Term Long-Term Senior
Outlook*Ba1Ba1
Income StatementBaa2Baa2
Balance SheetCaa2C
Leverage RatiosBa1C
Cash FlowBa1B1
Rates of Return and ProfitabilityB3Ba1

*Financial analysis is the process of evaluating a company's financial performance and position by neural network. It involves reviewing the company's financial statements, including the balance sheet, income statement, and cash flow statement, as well as other financial reports and documents.
How does neural network examine financial reports and understand financial state of the company?

Prediction Confidence Score

Trust metric by Neural Network: 93 out of 100 with 772 signals.

References

1. Breusch, T. S. A. R. Pagan (1979), "A simple test for heteroskedasticity and random coefficient variation," Econometrica, 47, 1287–1294.
2. Zubizarreta JR. 2015. Stable weights that balance covariates for estimation with incomplete outcome data. J. Am. Stat. Assoc. 110:910–22
3. E. Altman, K. Avrachenkov, and R. N ́u ̃nez-Queija. Perturbation analysis for denumerable Markov chains with application to queueing models. Advances in Applied Probability, pages 839–853, 2004
4. Meinshausen N. 2007. Relaxed lasso. Comput. Stat. Data Anal. 52:374–93
5. P. Artzner, F. Delbaen, J. Eber, and D. Heath. Coherent measures of risk. Journal of Mathematical Finance, 9(3):203–228, 1999
6. K. Boda and J. Filar. Time consistent dynamic risk measures. Mathematical Methods of Operations Research, 63(1):169–186, 2006
7. Bottou L. 1998. Online learning and stochastic approximations. In On-Line Learning in Neural Networks, ed. D Saad, pp. 9–42. New York: ACM
Frequently Asked QuestionsQ: What is the prediction methodology for GWR:TSX stock?
A: GWR:TSX stock prediction methodology: We evaluate the prediction models Statistical Inference (ML) and Multiple Regression
Q: Is GWR:TSX stock a buy or sell?
A: The dominant strategy among neural network is to Buy GWR:TSX Stock.
Q: Is Global Water Resources, Inc. stock a good investment?
A: The consensus rating for Global Water Resources, Inc. is Buy and is assigned short-term Ba1 & long-term Ba1 estimated rating.
Q: What is the consensus rating of GWR:TSX stock?
A: The consensus rating for GWR:TSX is Buy.
Q: What is the prediction period for GWR:TSX stock?
A: The prediction period for GWR:TSX is (n+8 weeks)


text:
# Exploiting time varying B field to employ vacuum acceleratio

Tags:
1. Aug 7, 2015

### BiGyElLoWhAt

So I've been wanting to build a particle accelerator for a while, and have kind of been brain storming ideas to make it work. I've been recently trying to figure out how to get the actual acceleration to happen.

I have a few ideas, but the one that I like (assuming it's possible) is using a time varying magnetic field to create a circular electric field. Is it possible to create this without a conductor to house the charges?

Also, what would be a reasonable db/dt to expect to get without emplying superconductors if I use electro magnets? Would it be better to use natural magnets? Perhaps rig up a device that drops natural magnets through the loop. If they are transported far enough away before bringing them back up, then I should be able to get a good amount of inductance going on, assuming charges are the only thing necessary to experience said induction, and not that they lie on a conductor.

Really the only other idea that would be coming to mind would be to use conductive loops wrapped around the vacuum tube (chamber) and switch the voltages on a somewhat periodic basis, but this would have to vary as the (likely) electrons or (hopefully) protons accelerate. I'm not sure how to measure that, or calculate it accurately enough to be able to switch the voltages and keep everything accelerating.

Thoughts? Ideas? Critisisms?
Thanks.

2. Aug 7, 2015

### Staff: Mentor

This is the basic idea of the betatron. Have you read up on those yet?

3. Aug 7, 2015

### DEvens

The Maxwell's equation with time derive of $B$ is $\nabla \times E = - \frac{\partial B}{\partial t}$.

"The voltage accumulated around a closed circuit is proportional to the time rate of change of the magnetic flux it encloses."

So, you can get voltage round a loop. But you cannot keep it there indefinitely. Eventually you reach the extreme of $B$ that you can achieve with whatever equipment you have. Then the $E$ field has to drop off. If you have an oscillating $B$ field of some kind, then you get an oscillating $E$ field. Your charged particles go one way for a while, then turn around and go the other way. Assuming they don't smack into the wall at some point.

Also, if you hope to get the charged particles to follow the circular path, then you need something to curve them into that path. Usually that means a magnetic field perpendicular to the path of travel. Otherwise, the force is going to fling the particles out of the loop. If the magnetic field is varying like crazy, then the radius of the orbits is also.

So you have a real balancing act here. You need to have the $B$ field increasing at exactly the right rate so that it provides exactly enough force to keep the charged particles going around the loop. Possibly you can have some extra "play room" if you have a range of radii that are acceptable, by having a disk shaped area for the particles to go round in. Possibly you can do some balancing of the strength of the $B$ field as a function of radius.

Eventually you reach whatever limit on $B$ your equipment enforces. And then you can't go any faster.

4. Aug 7, 2015

### BiGyElLoWhAt

No I have not. I might have to look into it. Do you think it's feasible from a diy point of view?

That was the whole reason I thought about physically dropping natural magnets through the loop, then moving them to the outside, bringing them back up, and dropping them again.

As far as stabalization goes, from what I've seen it's best to use 4 magnet per group and stabalize 1 axis per group. Using a magnet on opposite sides of the tube will create a net zero B field in the center and a B field around it that when mixed with the moving charge acts as a central force. I'm not really concerned with that. Just slap some magnets around the outside and point the north pole in. Alternate axes for centralization. My main concern is the actual acceleration part of it.

5. Aug 7, 2015

6. Aug 7, 2015

### BiGyElLoWhAt

With regards to the betatron, they're talking about pulsed magnets? It seems to me that the momentum imparted would have a time average of zero, so how does that work?

7. Aug 7, 2015

8. Aug 7, 2015

### Staff: Mentor

Please note that PF's policy is to not allow discussion of the actual construction of such a device because of the potential dangers (vacuum, high voltages, etc.). Discussion of the theory of operation is OK.

9. Aug 8, 2015

Noted.


text:
# Prove that ${1\over2}<{1\over1001}+…+{1\over2000}<1$

Prove that ${1\over2}<{1\over1001}+......+{1\over2000}<1$

Can it be proved by langrange's mean value theorem or by convert it into a Riemann sum?

• Hint: You're vastly overkilling it. – Git Gud Jun 6 '14 at 11:39
• Why proving that a finite sum is in $(0.5,1)$? Just calculate the value and you're done. – Thekwasti Jun 6 '14 at 11:42
• – OBDA Jun 6 '14 at 11:56

$$\frac{1}{2}=\frac{1000}{2000}<\underbrace{\frac{1}{1001}+\cdots+\frac{1}{2000}}_{1000 \text{ terms}}<\frac{1000}{1000}=1$$


text:
# zbMATH — the first resource for mathematics

Approximation of waves in composite media. (English) Zbl 0682.73028
The mathematics of finite elements and applications, VI, MAFELAP 1987, Proc. 6th Conf., Uxbridge/UK 1987, 55-74 (1988).
[For the entire collection see Zbl 0652.00018.]
We are concerned with the numerical simulation of waves in a composite, isotropic medium $$\Omega \subset {\mathbb{R}}^ 2$$ consisting of an elastic solid subdomain $$\Omega_ s$$ surrounding a fluid-filled porous subdomain $$\Omega_ p$$. The elastic wave equations and Biot’s low frequency dynamic equations are used to describe the propagation of waves in $$\Omega_ s$$ and $$\Omega_ p$$, respectively. Boundary conditions allowing conservation of energy flux are applied at the contact surface between $$\Omega_ s$$ and $$\Omega_ p$$, while absorbing boundary conditions are employed at the artificial boundaries of the model and a stress-free condition is used at the free surface of the domain.
In this paper we show the results obtained by the implementation of a finite element technique and compare the results with the corresponding ones for an inhomogeneous solid in which the porous medium is replaced by an elastic solid with physical properties partially equivalent to those of the porous medium in a sense to be explained later.
In §2 we describe the composite model and then derive a weak form of it. Then we discuss the finite element spaces used to discretize the problem and introduce the finite element procedure. Results concerning the existence and the uniqueness of the solution of the differential and finite element systems and the convergence of the approximate solution to that of the differential system are summarized and referenced. In §3 we describe the experiments performed and present the results of these experiments. Conclusions and some suggestions of continued research on the subject are also included.

##### MSC:
 74J99 Waves in solid mechanics 74S05 Finite element methods applied to problems in solid mechanics 74E30 Composite and mixture properties 76S05 Flows in porous media; filtration; seepage
Zbl 0652.00018