# Tagged Questions

Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

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### Radius of convergence of integral series; problem with limsup

Let $\sum c_k x^k$ be a power series with radius of convergence $R$. Then the integral series $$\sum_{k=0}^\infty \frac{c_k}{k + 1}x^{k+1}$$ also has radius of convergence $R$. I'm reading Real ...
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### Find the radius of convergence as well as the interval of convergence:

I've done a bunch of these and was successful, but this one is proving to be troubling. I have no idea how to handle that n*sqrt(n) at the bottom. I ended up with some jank shit like this lol: Lim ...
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### Find the relative width of a guitar fret

There is an equation to find the position of a fret on a guitar fretboard, given the length of a string is given by \begin{eqnarray} d = s – \frac{s}{2 ^ {(n / 12)}}, \end{eqnarray} where $d$ is the ...
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### Why is $\operatorname{Sech}(x)$ Taylor series divergent past $\pi/2$?

Someone asked a question here about why the Taylor series of $\log(1+x)$ diverges: Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$? I have a similar question: why ...
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### Can the series $\sum_{n=0}^\infty \frac{(-1)^n}{\sqrt{2^n n!}} x^{n}$ be summed? [closed]

Can the following series $$\sum_{n=0}^\infty \frac{(-1)^n}{\sqrt{2^n n!}} x^{n}$$ be summed?
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### Find the Taylor-series expansion of a square of a rational function of a complex variable

I've been trying to find the Taylor-series expansion of the following function: $$f(z)=\left ( \frac{1+z}{1-z} \right )^2$$ az the origin : Z0 = 0. also I would like to find the region of ...
### What is the interval of convergence of : $\sum_{n=1}^\infty\frac{n^n}{n!}x^n$?
$x+ \frac{2^2x^2}{2!}+ \frac{3^3x^3}{3!}+ \frac{4^4x^4}{4!}+...$ Possible answers- 1.($0,1/e$) 2.(1/e, $\infty$) 3.(2/e, 3/e) 4.(3/e, 4/e) ...