# Tagged Questions

4answers
128 views

### Are Taylor series and power series the same “thing”?

I was just wondering in the lingo of Mathematics, are these two "ideas" the same? I know we have Taylor series, and their specialisation the Maclaurin series, but are power series a more general ...
2answers
58 views

### How does one get the Bernoulli numbers via the generating function?

Here is the definition: Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$ I've tried to naively expand $\frac{x}{e^x-1}$ around ...
1answer
33 views

### Points around which one expands and the radiuses of convergence

I'm trying to make sense of the following passage: Let $f(x)=\frac{1}{x+1}$ and $R_0$ the radius of convergence of the Taylor series of $f$ around $x_0=0$, analogously: $R_1$ — around ...
2answers
45 views

### Analytic functions of a real variable which do not extend to an open complex neighborhood

Do such functions exist? If not, is it appropriate to think of real analytic functions as "slices" of holomorphic functions?
1answer
55 views

### Why do power series converge to a function symmetrically?

Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$? The selected answer to the above question says that for a a power series, the interval of convergence for the ...
3answers
32 views

### Demonstrate that $\frac{1}{e^e} - 1 + e - \frac{e^2}{2} + \frac{e^3}{6}\ge 0$

How do I prove the inequality? $$\frac{1}{e^e} - 1 + e - \frac{e^2}{2} + \frac{e^3}{6} \geq 0$$ I can see that \$e^e = \sum_{k=0}^{\infty} \frac{e^k}{k!} = 1 + e + \frac{e^2}{2} + \frac{e^3}{6}+\dots ...
2answers
46 views

### Taylor series of $f(x^2)$

If you know the taylor series for $f(x)$ can you find the taylor series for $f(x^2)$ by letting $x = x^2$? The taylor series in question is $\cos(x^2)$ I know the taylor series for $\cos(x)$ is ...
1answer
48 views

### Why do Maclaurin series approximate a function for negative domain values?

A common analogy used as an intuitive explanation for a Maclaurin series is that of a car. If you know the position, velocity, acceleration, jerk etc. of a car at time zero, you are able to predict ...
0answers
142 views

### power series of arcsin(x) centered at x = 0

I am trying to prove that the Taylor expansion of $\arcsin(x) = \sum\limits_{n=0}^\infty \cfrac{(2n!)x^{2n+1}}{(2^nn!)^2(2n+1)}$. Sorry about the notation, I'm not sure what syntax to use. S stands ...
1answer
64 views

### Adding two Power series or Maclaurin sums together and their radius of convergence

Say you have two power series. One of them has ROC of 2, and the other one has an ROC of 4. If you add the two series together is the ROC ALWAYS the lesser ROC? It seems to be a trend I've noticed, ...
0answers
40 views

### Taylors Inequality to evaluate $f(x) = x\sin(x)$ when $a = 0$ and $-1\le x\le1$

Trying to calculate the error of this function when you use a Taylor expansion to degree 4. I keep getting $.039$ when the answer in the back of the book is $.042$. I take the fifth derivative of ...
2answers
32 views

### Lagrange remainder to approximate $3^{2.1}$ less than 0.1

How do I solve this problem: Use the appropriate Taylor polynomial $P_n(x,c)$ to estimate $3^{2.1}$ with error less than $0.1$, given $\ln 3$ is about $1.099$. I understand that the remainder ...
1answer
41 views

### Find taylor polynomial that approximates e^x with accuracy at least 1.

Find Taylor polynomial at $x=0$ which approximates $e^x$ with accuracy at least $1$ for each $x \in [-2,2]$. I dont undestand these questions that involve the $n^{th}$ remainder. I know I need to ...
2answers
113 views

### Confused by Laurent series

A typical problem related to Laurent series is this: For the function $\frac 1{(z-1)(z-2)}$, find the Laurent series expansion in the following regions: \$\\(a) |z|<1, \\ (b) 1<|z|<2, ...
1answer
50 views

### Confused over analytic functions, point convergence of power series

It is well-known that a power series sums to a function that is analytic at every point inside its circle of convergence and that conversely, if a function is analytic on an open disc then its Taylor ...
2answers
102 views

### How do I obtain the Laurent series for $f(z)=\frac 1{\cos(z^4)-1}$ about $0$?

I know that $$\cos(z^4)-1=-\frac{z^8}{2!}+\frac{z^{16}}{4!}+...$$ but how do I take the reciprocal of this series (please do not use little-o notation)? Or are there better methods to obtain the ...
2answers
44 views

### Solving limit by substituting a power series

I dont understand why I am getting 2 and the textbook says it is -2. $$\lim_{x\to 0} \frac{1-e^x}{\sqrt{1+x}-1}$$ I subbed the power series for $e^x$ and $(1+x)^{1/2}$ then got rid of the $1$ on top ...
3answers
46 views

### Precise differences in meaning of Power Series, Taylor Series

Being an physicist/artist, not a real mathematician, I often toss around the terms "Taylor Series" and "Power Series" without any concern. Are these terms be considered interchangeable by ...
2answers
88 views

### Finding the Maclaurin series

Find the Maclaurin series for $f(x)=(x^2+4)e^{2x}$ and use it to calculate the 1000th derivative of $f(x)$ at $x=0$. Is it possible to just find the Maclaurin series for $e^{2x}$ and then multiply it ...
1answer
63 views

### Exponential as power series

Is there a function that does not depend on $a$ such that $\sum_{x=1}^\infty \frac{a^x}{x!}f(x) = \mathrm e^{-a}$? Just to be clear, the summation starting from 1 is intentional, otherwise the ...
2answers
144 views

### Proof of the “Radius of Convergence Theorem”

I can't figure out how it is valid to invoke the Absolute Convergence Theorem, whose hypothesis is "Let the power series have radius of convergence R", to establish case c of the Radius of Convergence ...
1answer
48 views

### Taylor series representation of a function.

I'm working on expressing the function $f(x)=\frac{6}{x}$ as a taylor series about $-4$. I've got the general idea, but I'm not quite there yet. I've come up with the equation ...
1answer
137 views

### Laurent Series and Taylor Series

I am trying to find the Laurent series of $\dfrac{1}{(1+x)^3}$; would this be the same as finding the Maclaurin series for the same function?
1answer
109 views

### Taylor / Maclaurin series expansion origin. [closed]

Soo we all know Taylor series expansion formula for expansion around expansion point $A(a,f(a))$: f(x) \approx \underbrace{f(a)}_{1st~term} + \underbrace{\frac{f'(a)\, (x-a)}{1!}}_{2nd~term}+ ...
1answer
102 views

### Laurent series of $$g(z)=\frac{z^n+z^{-n}}{z^2-(a+\frac{1}{a})z+1}=?$$

How to find Laurent series of g(z) ? $$g(z)=\frac{z^n+z^{-n}}{z^2-(a+\frac{1}{a})z+1} \hspace{10mm} \begin{cases} n \in N \\ 0<a<1 \end{cases}$$ answer is :  ...
1answer
91 views

### Differentiating power series

Consider the power series $$\sum_{n=0}^\infty{\frac{x^{2n}}{(2n)!}}$$ From this, it follows that its sum defines an infinitely differentiable function $f$, given by ...
2answers
393 views

### Find the Taylor Series for $f(x)$ centered at a given value $a$

$$f(x) = \frac{6}{x}\,\, \mathrm{at}\,\, a = -4 .$$ Assume that $f$ has a power series expansion. Do not show that $R_n(x) -> 0$ I took the derivatives of f(x): $$f(x) = 6/x$$ $$f'(x) = -6/x^2$$ ...
0answers
148 views

### domain of convergence of a multivariable taylor series

consider the rational function : $$f(x,z)=\frac{z}{x^{z}-1}$$ $x\in \mathbb{R}^{+}/[0,1]\;\;$, $z\in \mathbb{C}\;\;$ .We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type ...
2answers
142 views

### Why do we need Taylor polynomials?

This question doubles as "Is my understanding of what a Taylor polynomial is for, correct?" but In order to write out a Taylor polynomial for a function, which we will use to approximate said function ...
2answers
244 views

### Why is Taylor series expansion for $1/(1-x)$ valid only for $x \in (-1, 1)$?

After finding an expansion of $$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots$$ a quick test of various values for $x$ reveals that this expansion is not valid for $\forall x \in \mathbb{R}-\{1\}$. ...
1answer
114 views

### Asymptotic of Taylor series

Let $f(x)$ and $g(x)$ be two Taylor series such that: $$f(x)= \sum_{n=0}^{\infty}(-1)^{n} a(n) x^{n}$$ and $$g(x)= \sum_{n=0}^{\infty} b(n) x^{n}$$, for $a(n) >0$ and $b(n) > 0$. My ...
2answers
94 views

### Problem regarding infinite sum of remainders.

Before here @math.SE there was a question regarding a problem on a maths magazine. I decided to look at the link provided, and one problem proposed was (if I'm not recalling this wrongly): Find ...
1answer
195 views

### Power Series Definition

What does it mean for a series to be centered around a number? I'm taking complex analysis and am suddenly very confused. I didn't have this explanation, or proof of taylor and power series in ...
4answers
150 views

### Formula for calculating $\sum_{n=0}^{m}nr^n$

I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived. For example, when r = 2, the formula is given by: $\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$ ...
4answers
127 views

### Infinite series expansion of $e^{-x}\cos(x)$

Establish an infinite series expansion for the function $y=e^{-x}\cos(x)$ from just the known series expansions of $e^x$ and $\cos(x)$. Include terms up to the sixth power. I know that the ...
3answers
294 views

### A deceiving Taylor series

When we try to expand  \begin{align} f:&\mathbb R \to \mathbb R\\ &x \mapsto \begin{cases} \mathrm e^{-\large\frac 1{x^2}} &\Leftarrow x\neq 0\\ 0 &\Leftarrow x=0 ...
2answers
374 views

### Using Taylor series expansion as a bound

I have a function $f(x)$ that has convergent Taylor series expansion around $x=0$ in the following form: ...
3answers
176 views

### Taylor series for different points… how do they look?

I can't understand what it means to do the Taylor series at the point $a$. The best way would be showing me how it looks for different $a$ on a graph. Do I find those graphs on the Internet?
1answer
148 views

### Maclaurin series of $\frac{1}{1+x^2}$

I'm stumped here. I''m supposed to find the Maclaurin series of $\frac1{1+x^2}$, but I'm not sure what to do. I know the general idea: find $\displaystyle\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$. ...
1answer
157 views

### Really basic question about the Taylor expansion of a CDF

I am sorry for such a basic question... but I want to try to do a Taylor expansion on my function, which is a CDF defined over 0-1. However, when I expand around 0, which is what I read is typical, ...
1answer
123 views

### A question about the product of two series

Given two power series, $$f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}$$ and $$g(x)=\sum_{n=0}^{\infty}b_{n}x^{n}.$$ It is easy to form their product $$f(x)g(x)=\sum_{n=0}^{\infty}c_{n}x^{n}$$ where ...
2answers
959 views

### How to use the Lagrange's remainder to prove that log(1+x) = sum(…)?

Using Lagrange's remainder, I have to prove that: $\log(1+x) = \sum\limits_{n=1}^\infty (-1)^{n+1} \cdot \frac{x^n}{n}, \; \forall |x| < 1$ I am not quite sure how to do this. I started with the ...
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