# 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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### Deriving power series for $\sin x$ without using Taylor's Theorem or $\exp z$

Starting with defining $(\cos t, \sin t)$ from the unit circle, is it possible to derive the power series for $\sin(t)$: $$\sin t = t - \frac{t^3}{3!} + \frac{t^5}{5!} - \dots$$ Note: I will be ...
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### Taylor series for $\tanh(z)$

Find the Taylor series of $\tanh(z)$ around $z_0=0$. $z$ is a complex variable. I can use all the basic series as facts like the $\cosh$ and $\sinh$ series. I know how to calculate the series ...
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### The $n=0$ term in the power series $\sum_{n=0}^\infty a_n x^n$

This question is about the definition and notation for the $0^{th}$ term of the power series: $$\sum_{n=0}^{\infty} a_n x^n$$ There are two possible ways to interpret this term: 1) It is just a ...
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### Explicit expression of a given power series

Let us have a look to the power series of the form $$\sum_{n=0}^{\infty}{\frac{1}{n+2}x^n},\ \ \ x\in\mathbb{R}$$ I want to find an explicit expression of this power series. I think one have to us ...
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### Convergence of $\sum_{n=1}^\infty \frac{n!}{n^n} x^n$

I'm trying find out where $\sum_{n=1}^\infty \frac{n!}{n^n} x^n$ converges. First I found that the radius of convergence is $R=e$, but after that I had difficulty testing convergence at $x=\pm e$. I'...
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### Radius of Convergence on Power Series Help

I am struggling to find the radii of convergence of the following two series: $$\sum_{n}n^{\cos(n)}z^n$$ $$\sum_{n}(2^{-n} + 3^{-n})z^n$$ Here I tried using ratio test and lim sup, but didn't ...
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### Definition of an inverse-powerseries

Let $t(q)=\sum_{n=0}^{\infty}t_n q^n$ be a complex powerseries convergent for all $|q|<1$. Assume $t_0=0$ and $t_1\neq0$. Not it says Let $q(t)$ be the local inverse of $t(q)$ with $q(0)=0$. ...
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### Power series expansion of $z\mapsto \frac{\mathrm e^{z}}{1-tz}$ and $z\mapsto \tan z$

Determine the power series expansion and radius of convergence of $z\mapsto \frac{\mathrm e^{z}}{1-tz}$ around $0$ with $t\in\mathbb C$. Determine the radius of convergence and the first three non-...
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### Expand a function to power series

I have the following function and i try to expand it to a power series - $$F(x) = \frac{2x}{(x^2+1)^2}$$ around $X = 0$ I tried to substitute $t = -x^2$ and got stuck. I would like to get some help ...
I encountered the following problem: $$\lim_{x\to 0} \frac{x-\ln(1+x)}{x-\arctan x}$$ I expanded $\arctan x$ in the denominator up to the fifth term and get the following:  x - \left(x - \...