# Tagged Questions

2answers
33 views

### Checking $\displaystyle \int_{0}^{\infty} {\frac{\sin^2x}{\sqrt[3]{x^7 + 1}} dx}$ for convergence

Given $\displaystyle\int_{0}^{\infty} {\frac{\sin^2x}{\sqrt[3]{x^7 + 1}} dx}$, prove that it converges. So of course, I said: We have to calculate \$\displaystyle \lim_{b \to \infty} ...
2answers
18 views

### Develop the next function:$f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0)$

Develop the next function:$\displaystyle f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0).$ For the first part: \$\displaystyle\frac ...
7answers
124 views

### How to prove that $1/n!$ is less than $1/n^2$?

I want to prove $$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test. How to prove ...
1answer
16 views

### Absolute convergence.

Determine if absolutely convergent or not; Justify. $$\sum_{n=1}^\infty (-1)^n n^2 3^{1-n} x^n \text{ s.t }|x|<3$$ if we take the abs value of $(-1)^n$ we are left with $n^{2} 3^{1-n} x^{n}$ now ...
3answers
28 views

### behavior of a sequence

Imaging a sequence $a_{k} \in \Omega$ with $\Omega \subset \Bbb{R}$ closed, $\lim\limits_{k \to \infty} \| a_{k+1} - a_{k} \| = 0$. My Professor said that because of this the sequence would ...
2answers
32 views

### What's the radius of convergence of the next sum: $\sum_{n=0}^\infty (\int_o^n\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt)x^n$

What's the radius of convergence of the next sum: $$\sum_{n=0}^\infty \left(\int_0^n\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt\right)x^n$$ I know that $$\int_0^\infty\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt$$ does ...
1answer
49 views

### Conditions for taking a limit into an infinite sum

Suppose $f\left(x\right)={\displaystyle \sum_{n=0}^{\infty}g_{n}\left(x\right)}$ under what conditions is it true that: \lim_{x\to c}f\left(x\right)={\displaystyle \sum_{n=0}^{\infty}\lim_{x\to ...
1answer
36 views

### Radius of convergence of a power series with Bernoulli numbers

Say, we use the definition: Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$ and then derive power series representations of the ...
1answer
30 views

### Calculate (& determine if converges): $\sum_{n=1}^\infty (-1)^n\frac 1n(0.5)^n$

Calculate (& determine if converges): $\sum_{n=1}^\infty (-1)^n\frac 1n(0.5)^n$ The above is a specific equation for $x_0=0.5$, but from here i'm pretty much stuck.
1answer
67 views

### Radius of convergence of: $\sum_{n=1}^\infty\frac {x^{n!}}{n!}$?

What the Radius of convergence of: $\sum\limits_{n=1}^\infty\frac {x^{n!}}{n!}$? So far I tried finding the ...
2answers
63 views

### A question on the convergence of a Taylor series of some prominent function

The function $f:\mathbb{R}\to\mathbb{R}$ defined by $$f(x)=\begin{cases}e^{-\frac{1}{x^2}} &if &x\neq 0\\ 0 & else \end{cases}$$ is a prominent example of a function whose Taylor series ...
3answers
91 views

### How can I prove that $\int_1^\infty \left\lvert\frac{\sin x}{x}\right\rvert dx$ diverges?

I know a start could be to try and prove that $\int_1^\infty \frac{\sin^2x}{x} dx$ diverges since $\frac{\sin^2x}{x} \le \left\lvert\frac{\sin x}{x}\right\rvert$ in this interval, but I wouldn't know ...
1answer
45 views

### Radius of convergence of $\sum_{n=1}^{\infty} { (n \sin{\frac{1}{n}})^{n} x^n }$

We need to calculate the radius of convergence $R$ of: $$\sum_{n=1}^{\infty} {\left(n \sin{\frac{1}{n}}\right)^{n} x^n }.$$ Here's what I did:  \lim_{n\to\infty} { \left| ...
3answers
78 views

### Radius of convergence for $\sum_{n=1}^{\infty} { \frac{1}{n!} \cdot x^{n!}}$

How do we find the radius of convergence $R$ of this power sum? $$\sum_{n=1}^{\infty} { \frac{1}{n!} \cdot x^{n!}}$$ How do we handle the $n!$ as the power of $x$?
2answers
43 views

### Finding the radius of convergence and what does it mean

We have just started learning this and I do not really understand it fully. Given: $$\sum_{n=1}^{\infty} {(\pi^n + n + 1) \cdot x^n}$$ We should check what is the radius of convergence $R$ and what ...
3answers
68 views

### Show that $(x_n-y_n)$ converges to $x-y$.

Given $(x_n)$ and $(y_n)$ are sequences of real number which converge to $x$ and $y$ respectively. Show that $(x_n-y_n)$ converges to $x-y$. If it's asking about $(x_n+y_n)$. I know that I can ...
1answer
29 views

### Study the convergence of this sequence of functions

I have the following sequence of function: $$f_n(\lambda)=\bigg[\alpha-i\bigg(\lambda+\frac{1}{n}\bigg)\bigg]^{-1}-\bigg[\alpha-i\lambda\bigg]^{-1},\,\,\,\alpha\neq 0$$ and I have to study its ...
1answer
57 views

### Series of Functions - Pointwise and Uniform Convergence.

I was hoping for some help for the following questions. Prove that the series $\sum_{n=1}^\infty x^n(1-x)$ converges pointwise but not uniformly on $[0,1]$. Prove that the series \$\sum_{n=1}^\infty ...
3answers
66 views

### Show that the sequence $(x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition.

Show that the sequence $\displaystyle (x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition. I'm not familiar with proving divergent sequence. Do anyone have any des? ...
1answer
49 views

### Prove convergence of improper integral using change of variable.

This may be trivial, but I could use some help... Consider a real function $f: (0,1) \rightarrow \mathbb{R}$, continuous, positive, but not necessarily bounded. Let $g: [0,1] \rightarrow [0,1]$ be a ...
2answers
48 views

### Prove or disprove a result for a double sequence.

Suppose that a double sequence $\{a_{n,k}\}=\left\{\frac{1}{n^{\frac{k-1}{k}}}\right\}$. Prove or disprove $\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{k}a_{n,k}=0$.
2answers
88 views

### What is $\lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$?

How to solve the following limit question? $$\lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$$ Thanks a lot.
1answer
62 views

### When does $\lim\limits_{n\to\infty}\int_{b}^{a_n}f_n(x)dx=\lim\limits_{n\to\infty}\int_b^\infty f_n(x)dx$ hold?

Let $\{a_n\}\subset \mathbb{R}$ be sequence and $$f_n:[b,\infty)\longrightarrow \mathbb{R}, \qquad n=1,2,\dots .$$ Assume that $$\lim_{n\longrightarrow\infty}a_n=+\infty.$$ Obviously, from the ...
3answers
125 views

### Evaluating $\lim \limits_{n\to \infty} \left( n \int_{0}^{\frac \pi 2} 1-\sqrt [n]{\sin x} \,\mathrm dx \right)$

Evaluate the following limit: $$\lim \limits_{n\to \infty} \;\; n \int_{0}^{\frac \pi 2} \left(1-\sqrt [n]{\sin x} \right)\,\mathrm dx$$ I have done the problem . How I solved is First I ...
2answers
85 views

### Interval of convergence of $\sum_{n=4}^\infty x^n/n^5$

Find the interval of convergence $$\sum_{n=4}^\infty x^n/n^5$$ I'm lost here. My intuition was to use the ratio test. $$\lim_{n \to \infty} \frac{x^{n+1}}{(n+1)^5} \times \frac{n^5}{x^n}$$ ...
3answers
97 views

### How to solve $\sum_{k=2}^\infty {\frac{1}{k^2-1}}$

I'm using the integral test to determine if this series converges. From what I have so far it seems that it diverges, but according to wolfram alpha it converges. Where is my mistake? ...
5answers
135 views

### Convergence of a sequence $c_n$

Suppose that $(a_n)$ and $(b_n)$ be sequences such that $\lim (a_n)=0$ and $\displaystyle \lim \left( \sum_{i=1}^n b_i \right)$ exists. Define $c_n = a_1 b_n + a_2 b_{n-1} + \dots + a_n b_1$. Prove ...
1answer
189 views

### binomial expansion formula proof, bases on Lagrange form of Taylor series remainder

Another exercise from Bartle/Sherbert Introduction to Real Analysis book (this one is exercise 9.4.14): Use the Lagrange form of the remainder to justify the general Binomial Expansion ...
0answers
36 views

### Is this pointwise convergence sequence also uniform convergence?

$f_{n}$ and $f$ are continuous functions and $f_{n}\rightarrow f$ pointwise. Which of the following are correct? $\int _{0}^{x}F_{n}\left( t\right) dt\rightarrow\int _{0}^{x}F\left( t\right) dt$ ...
1answer
47 views

### convergence of complex series.

For what values ​​of $(a,b\in \mathbb{C})$ does this series converge or diverge? $\sum\frac{(k-a)^2}{(k-b)^3}$ if $a,b\in\mathbb{R}$ by the Limit comparison test (with $\sum\frac{1}{k}$ ) I know ...
3answers
67 views

### convergence of $\sum\frac{a_{n}}{n}$ if $\sum_{k=1}^{n}a_{k}\le M*n^{r}$ where $r<1$

Show that if the partial sums $s_{n}$ of the series $\sum_{k=1}^{\infty}a_{k}$ satisfy $|s_{n}|\le M*n^{r}$ for some $r<1$, then the series $\sum_{n=1}^{\infty}\frac{a_{n}}{n}$ converges.
2answers
30 views

### Radius of convergence in a series. Ratio test.

I am having a hard time with this question. $$\sum_{k=0}^{\infty} \frac{-(1)^k (4^k -3)x^{2k}}{k^4+3}$$ I used the ratio test and got stuck here: x^2 \lim_{k\to\infty} \frac ...
1answer
100 views

### Proof that $\sum\limits_{k=1}^n\frac{\sin(kx)}{k^2}$ convergences uniformly using the Cauchy criterion

I'd like to use the Cauchy criterion to show that $$f_n(x)=\sum\limits_{k=1}^n\frac{\sin(kx)}{k^2} \mbox{convergences uniformly}$$ Here is what I did: We want to show that \$\forall \, ...
3answers
89 views

### Does this series converge or diverge?

I have a series here, and I'm supposed to determine whether it converges or diverges. I've tried the different tests, but I can't quite get the answer. ...
1answer
49 views

### Composition Taylor Series

Is there any theorem that specifies when we are allowed to compose the taylor series of two functions? Does it have a name? Thanks.
1answer
31 views

### Two quick questions about convergence (in the context of pointwise vs. uniform convergence)

I found this example online: Let $\{f_{n}\}$ be the sequence of functions on $(0,\infty)$ defined by $f_{n}(x)=\frac{nx}{1+n^{2}x^{2}}$ .This function converges pointwise to zero. ...
3answers
64 views

### Representing Functions as Power Series

Rewrite $$f(x)=(1+x)/(1-x)^2$$ as a power series. Work thus far: I separated it into two parts: $$1/(1-x)^2 + x/(1-x)^2$$ I realize that the first expression is the derivative of $1/(1-x)$ and ...
1answer
29 views

### Specify conditions for $\alpha$ so that the iteration $x_{n+1} = x_n - \alpha f(x_n)$ converges to root of f.

Specify conditions on $\alpha$ so that the iterative process $x_{n+1} = x_n - \alpha f(x_n)$ converges to root of f if started with $x_0$ close to the root. It is suggested that the proof should ...
1answer
56 views

### Value of limsup i?

This is a part of my question. $\lim \sup \cos(n\pi/12)$ as n goes to infinity What is the value of this limit?
1answer
81 views

### Cauchy but not fast cauchy

As the title indicates, I am trying to find a Cauchy sequence that is not fast (or rapidly) Cauchy. Could anyone suggest something? A sequence $\{a_n\}_{n \in \Bbb N}$is termed fast (or rapidly) ...
3answers
55 views

### Convergence of definite integral

I have to find out the convergence of the next integral: $$\int^{\pi/2}_0{\frac{\ln(\sin(x))}{\sqrt{x}}}dx$$ Any help? Thanks
2answers
29 views

### What does $\sum_{n=0}^{\infty} (-1)^n .\frac{x^{2n+1}}{2n+1}$ converge to at x= 1 and x = -1

This is what I did. $\sum_{n=0}^{\infty} (-1)^n .\frac{1^{2n+1}}{2n+1}=\sum_{n=0}^{\infty}(-1)^n .\frac{1}{2n+1}$ Now I broke it up to positive and negative. \$\sum_{n=0}^{\infty}(-1)^n ...
1answer
89 views

### A question on uniform convergence

Let $f,g:[0,1]\rightarrow \mathbb{R}$ be continuous functions. Define $$\displaystyle x_{n}(t)=f(t)+ \int_{0}^{t}x_{n-1}(s)ds\quad 0\leq t\leq 1,\;n=1,2,\cdots ,$$ where $x_{0}(t)=g(t),0\leq t\leq 1$ ...
2answers
68 views

### Checking convergence of an improper integral

I did a quick search here but couldn't find a similar problem (it's probably out there somewhere...) I'm stuck with this rather simple improper integral: $\int_{1}^{\infty} \frac{1}{x^{\alpha}-1}dx$ ...
2answers
85 views

### Question about convergence of series if $\{n a_n\} \to 0$

This is part of Rudin's PMA Exercise 3.14 (d). If I understand correctly, it would be helpful to prove the following: Let $a_n$ be some sequence. Assume that $\lim_{n\to\infty} na_n = 0$. Prove ...
3answers
87 views

### Uniform convergence of a sequence of functions

Let $(u_n(x))$ be a sequence of functions in $(0,\infty)$ such that: $u_1(x)=x$, $u_{n+1}=\frac12\left(u_n(x)+\frac1{u_n(x)}\right)$ for $n\in\mathbb{N}$. Check if $u_n(x)$ converge unfiormly in ...
0answers
42 views

### Convergence of integrals over divergent parts

I'm wondering if it is possible for an integral which diverges in the limits $1$ to $\infty$ to converge in the limits from $0$ to $\infty$. And if so: how could I find this out? For example  ...
2answers
121 views

### $\sum\limits_{n=1}^\infty |a_n|$ converges implies $\sum\limits_{n=1}^\infty |a_n|^2$ converges? [duplicate]

Possible Duplicate: Prove that $\sum_{n=1}^{\infty}\ a_n^2$ is convergent if $\sum_{n=1}^{\infty}\ a_n$ is absolutely convergent If $\sum\limits_{n=1}^\infty |a_n|$ converges, the ...
1answer
38 views

### Convergence integral causal function

I have an exercise where there is the following given: $f$ is a causal function. $f$ is Laplace transformable:$\int_{0}^{\infty} f(t)e^{-zt} \, dt = L(z)$ with $Real(z)> -1$ I have to ...
1answer
59 views

### Why one comparison test and not the other

I'm studying integral comparison tests and I come to this one. $$f(x)=\int_{1}^{\infty}\frac{\sqrt{x}}{x^5+\sqrt[3]{x}}dx$$ The solution provided is to do \$\displaystyle ...
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 120, "mathjax_display_tex": 23, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9022445678710938, "perplexity_flag": "head"}