# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 $$\sum_{n=0}^\infty \dfrac{x^n + 2(-x)^n}{n!}$$ ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
### 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 ...