Questions tagged [power-series]

Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n (x-c)^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex.

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Formulas for $\pi$ of the form $2\sum_{k=0}^\infty\binom{2k}{k}\frac{a^{2k+1}+b^{2k+1}+c^{2k+1}}{4^k(2k+1)}$

Third edit: For those interested in the Sagemath-code to produce your own formula, given three natural numbers $x<y<z$, it can be found here. I am sharing those formulas in public domain, for ...
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Simplifying $\sum_{n=1}^{\infty}\frac{n\alpha^n}{(n-m)!(n+z)}(x-c)^{n-m}$

We would like to simplify the following summation $$\sum_{n=1}^{\infty}\frac{n\alpha^n}{(n-m)!(n+z)}(x-c)^{n-m}.$$ What we know is that $x, z, \alpha, c > 0$ and $z$ is integer. Here's what I did ...
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Power series as analitycs functions

I am searching how to prove that a power series is an analityc function because it is something that I have studied but without a prove.
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Suppose $\sum_{i = 1} a_i \sum_{i = 1} b_i < \infty$. If $\sum_{i = 1} a_i = \infty$, what does it say about $(b_i)$?

Let $(a_i), (b_i)$ be two non-negative sequence. Suppose $\sum_{i = 1} a_i \sum_{i = 1} b_i < \infty$. If $\sum_{i = 1} a_i = \infty$, what does it say about $(b_i)$? Does it necessarily mean that ...
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Power series interval of convergence $x^k$ multiplication/division

If we have a power series and we multiply it by $x^k$ where $k$ is fixed natural number, does the resulting series have the exact same interval of convergence? Also if we have power series starting ...
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Developping Gamma Function in power series

Let take the following function, which is similar to the Gamma Function: $$f(x) = \int_{0}^{\infty }t^{x^{2}-1}e^{-2t}dt$$ How could we develop this function in power series? I try to develop it ...
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Proving $\prod_{n=0}^{\infty}\left(1+\frac{x}{a^n}\right)=\sum_{n=0}^{\infty}\frac{(ax)^n}{\prod_{k=1}^{n}(a^k-1)}$

By trying to prove that Riemann's Zeta function is analytically expendable to the whole plane with one pole, I went aside and noticed this identity about formal power series (which are obviously ...
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Can $f(x)=\int_{0}^{\infty}t^{x^2+1}e^{-2t}dt$ be written as a power series in a neighbourhood of zero?

Let $f(x)=\int_{0}^{\infty}t^{x^2+1}e^{-2t}dt$. Can $f(x)$ be written as a power series in a neighbourhood of zero? In this case, what would its convergence radius be? Trying to solve this problem I ...
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Formal Taylor Series Expansion

In this paper, the authors consider how to convert a system of $N$ ODEs to a PDE where $N \rightarrow \infty$ in some appropriate sense. In what follows $u_{n+1}$, the solution of the $(n+1)$th ODE, ...
Consider the following iterative process. We start with the function having all $1$'s in its Taylor series expansion: $$f_0(x)=\frac1{1-x}=1+x+x^2+x^3+x^4+O\left(x^5\right).\tag1$$ Then, at each step ...
Continuation of an asymptotic series originally defined for $z>0$ to $z<0$
Consider the well-known asymptotic expansion of the $\Gamma%$-function: \Gamma(x)\sim\left(\frac{x}{e}\right)^x \sqrt{\frac{2 \pi }{x}}\left[1+\frac{1}{12 x}+\frac{1}{288 x^2}-\frac{139}{51840 x^3}-\...