0
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0answers
6 views

Range of convergence for Taylor's series gf given that of g and of f

Are the following 2 points correct? Let $D_f$ denote the maximal domain for which the Taylor's series of $f$ converges. 1) If $D_g = \mathbb{R}$, then $f$ converges $\implies gf$ converges. 2) On ...
1
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1answer
23 views

Range of convergence for Taylor's series (about 0) for e^(sin x)

Is there anything wrong with my method below? Also, is there an easier method? For $sin\,x = \sum^{\infty}_{k=0}\frac{(-1)^kx^{1+2k}}{(1+2k)!}$, $L_1 = \lim_{k\rightarrow\infty} \left| ...
1
vote
0answers
18 views

Expand a function in Maclaurin's series

Please help me expand this function in Maclaurin's series to find an interval of convergence. I tried to turn it into this expression: but it doesn't give me the result. Help me please it's ...
3
votes
2answers
65 views

Decide convergence of the series

Using Taylor expansion decide convergence of the series: $$\sum_{n=1}^{\infty}(e-(1+{{1}\over{n}})^n)^p = \sum_{n=1}^{\infty}a_n$$ I expanded $a_n$ like this $a_n = (e-(1+{{1}\over{n}})^n)^p = ...
0
votes
1answer
21 views

Calculating MacLaurin series for $\frac{1}{1-x^2}$

We have the M-series for $\frac{1}{1-x} = \sum_{n=0}^\infty x^n, \frac{1}{1+x} = \sum_{n=0}^\infty (-x)^n,$ and $\frac{1}{1-x^2} = \sum_{n=0}^\infty (x^2)^n$. I need to use the product of the first ...
0
votes
2answers
40 views

Taylor series expansion and the radius of convergence

Hello I have some problems concerning Taylor series. Given the function $$f(x)=e^{\sin{x}} $$ I concluded that the Taylor series expansion would be $$f(x) = ...
1
vote
1answer
40 views

Convergence question and degree of polynomial

I'm currently teaching myself power series and Taylor's theorem for complex analysis and I'm having trouble answering questions of the following form: $1)$ Suppose the power series ...
2
votes
0answers
80 views

Infinitely differentiable function with divergent Taylor series?

I'd greatly appreciate it if someone could provide examples of the following: 1) A infinitely differentiable function whose Taylor series does not converge to the function. 2) An infinitely ...
0
votes
1answer
34 views

Help with Taylor series problem

I am using maple to plot the graphs of e^e^x versus its truncated Taylor series around 0. For small values of x, the two graphs converge nicely, but once x<-3, my Taylor series loses control. Here ...
1
vote
0answers
43 views

Error Term of the Taylor series of cosh

I have the Taylor series of cosh $$\sum_{n=0}^\infty \frac {x^{2n}} {(2n)!}$$ and I know that this series converges for all x, but now I want to know if the series represents the function, in other ...
1
vote
1answer
94 views

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

I have to construct the Taylor series of $$\frac 1 {1+x^2}$$ around $0$ and $1$ and analyze the convergence in both cases. Also (but this is a consequence of the previous series) I have to construct ...
0
votes
0answers
82 views

Convergence of Taylor series of analytic function

Let $f(x)$ be analytic on $D= \{x \in \mathbb R^2: |x|< 1\}$. Then for $x_0 \in D$ there is an open set $U$ such that for all $x \in U$: $\sum_{n=0}^\infty a_n (x-x_0)^n = f(x)$, that is, the ...
2
votes
1answer
202 views

Taylor series convergence for $e^{-1/x^2}$

Consider the Taylor series for $e^{-1/x^2}$ around $0$: $$e^{-1/x^2}=1-\dfrac{1}{x^2}+\dfrac{1}{2!x^4}-\dfrac{1}{3!x^6}+\ldots$$ For which $x$ does the series on the right converge to $e^{-1/x^2}$?
2
votes
1answer
52 views

Laurent series, radii of convergence.

I'm working on the following exercise: Prove that a Laurent series \begin{align*} \sum_{n = -\infty}^\infty a_n(z-z_0)^n = \sum_{n = 0}^\infty a_n(z-z_0)^n + \sum_{n = 1}^\infty ...
4
votes
3answers
94 views

How to check the real analyticity of a function?

I recently learnt Taylor series in my class. I would like to know how is to possible to distinguish whether a function is real-analytic or not. First thing to check is if it is smooth. But how can I ...
2
votes
2answers
30 views

Does |Taylor Series of $f$ - $f$| Converge Monotonically to $0$?

Suppose that $T_n(x)$ be the sum of the first $n$ terms of the Taylor series of $f$ centered at $a$, and $\lim_{n\to \infty} T_n(b)=f(b)$. Is the difference $|T_n(b)-f(b)|$ decrease monotonically? ...
1
vote
0answers
50 views

Radius of convergence for Taylor series?!

Given is: $f(x) = \frac{\sin x}{x} $ I need the Taylor series in $a = 0$, so: $$T(x,0) = \frac{1}{x} \sum_{n=0}^\infty ((-1)^n* \frac{x^{2n+1}}{(2n+1)!} ) = \sum_{n=0}^\infty (-1)^n * ...
3
votes
1answer
114 views

Maclaurin series for $e^x +2e^{-x}$

I'm currently stuck on the question regarding the Maclaurin series for $e^x +2e^{-x}$ I've found that the power series representation for it is $$\sum_{n=0}^\infty \dfrac{x^n + 2(-x)^n}{n!}$$ ...
0
votes
1answer
69 views

Show the Newton method converges to 0 quadratically?

Using taylor series, show that if $x_n$ converges to a root, $f(x_n)$ usually converges to 0 quadratically. I reached a point I think I need to show that $\lim_{x\to \infty} ...
2
votes
1answer
162 views

Is exponential function analytic over all complex numbers

In my textbook, I find a text where it says $e^z$ is analytic everywhere (in complex plane). Is it true? If so, what is the proof? I approached using maclaurin series, which gives $e^z= ...
1
vote
1answer
49 views

MacLaurin powerseries and interval of convergence

Given the function $f(x) = 5/(6*x^2-x-1)$, (a) Expand into MacLaurin powerseries the function $f$ up to order $3$. (b) Find the interval of convergence of it. (a) I will use the type of ...
0
votes
0answers
46 views

funcitonal series convergence… SOS… [duplicate]

Is it possible to show that $f(x) = \sum_{k=0}^\infty (-1)^k \frac{x^k \sqrt{k}}{k!}$ converges even when $x\rightarrow \infty$ ? i.e. As a function of x, does $\lim_{x\to\infty} f(x)$ exist or at ...
1
vote
1answer
116 views

Integral remainder converges to 0

I want to show that $\displaystyle \log(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n}$ for $-1<x\leq1$ i want to show it with the integral remainder of the taylor series that gave me: ...
2
votes
2answers
148 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 ...
2
votes
1answer
280 views

Radius of convergence of Maclaurin series for $\frac1{\sin z}-1/z+\frac{2z}{z^2-\pi^2}$

What is the radius of convergence of the Taylor series about $z=0$ for $h(z)=\frac1{\sin z}-1/z+\frac{2z}{z^2-\pi^2}$? Here's a plot ...
2
votes
2answers
464 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, ...
2
votes
1answer
185 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 ...
3
votes
2answers
147 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 ...
1
vote
1answer
223 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.
1
vote
1answer
121 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 ...
1
vote
2answers
295 views

Stirling's Formula - Comparison Test Method

The following question concerns the convergence of Stirling's Approximation for $n!$ I have $r_n = \frac{\sqrt{n}}{n!}(\frac{n}{e})^n$. I have expressed ...
1
vote
3answers
1k views

radius of convergence of $1/(1+x^2)$

Which is the radius of convergence of Taylor series of $f(x)=\frac{1}{1+x^2}$? I am unable to write down the analitycal expression of all its derivatives. Why is it finite even if $f(x)<\infty$ ...
1
vote
1answer
142 views

Hessian gives a worse approximation of a multivariate function

I have a real, smooth, multivariate (with 10 variables or many more) function, for which I have the exact Jacobian and Hessian. It turns out that unless the norm of the increment of the function is ...
5
votes
1answer
834 views

Approximating $\arctan x$ for large $|x|$

I would like to know if there is reasonably fast converging method for computing large arguments of arctan. Until now I've came across Taylor series that converges only on interval $(-1,1)$ and for ...
1
vote
1answer
212 views

lower bound for the radius of convergence of Taylor series [duplicate]

Possible Duplicate: Radius of convergence of power series In 4 years of studying physics I came across a lot of Taylor series. All of them converged in a disc with a radius equal to the ...
4
votes
4answers
418 views

Non-converging Taylor series of $1/(1-x)$

From my own calculations with Maclaurin series and double-checking online, I get the result that: $$ {1 \over 1 - x} = \sum_{n=0}^\infty x^n $$ This seems to be true for $ -1 \lt x \lt 1 $, but ...
1
vote
0answers
232 views

Convergence of the Taylor series for the sine function

I would like to know if the Taylor series for the sine function, $$\sin{x} = x -\frac{x^3}{3!} + \frac{x^5}{5!} - \cdots,$$ is convergent if the argument of the function, $x$, is expressed in ...
0
votes
2answers
150 views

Domain of convergence of $f^{-1}: \mathbb R ^N \mapsto \mathbb R^N$ taylor series

In another question, I ask about the topology of the singular manifold of the Jacobian. What i want to ask in here is about the radius of convergence of a Taylor series expansion of the inverse ...
3
votes
1answer
295 views

Laurent series for an infinite sum of functions

I have an infinite sum of analytic functions that is guaranteed to converge for every $x$, except for $x=0$: \begin{equation} g(x) = \sum_{n=1}^\infty f_n (x) \end{equation} I want to expand the ...
19
votes
1answer
263 views

Are there always singularities at the edge of a disk of convergence?

Take a function that is analytic at 0 and consider its Maclaurin Series. Here are some examples I'll refer to: $$\frac{1}{1-x} =\sum_{n=0}^\infty x^n$$ $$\frac{1}{1+x^2} ...
6
votes
6answers
2k views

Don't understand why this binomial expansion is not valid for x > 1

today I'm studying binomial expansion and I'm a little confused about when certain expressions are valid. E.g. take this solution from my textbook: I understand that $(1-x)^{-1}$ has an infinite ...