13
votes
1answer
2k views

The unreasonable effectiveness of Pade approximation

I am trying to get an intuitive feel for why the Pade approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence. But what I can't ...
8
votes
4answers
658 views

What is the theoretical interest of finding closed-form sols. of infinite series?

Hi, I was reading this when I came across Gourevitch's conjecture. My understanding is that solutions to these series are of practical interest. If one encounters such a series, being able to solve ...
4
votes
2answers
282 views

Evaluating a limit similar to the Euler constant

In the course of studying a certain complex-valued functional equation, I have had a need to evaluate the following limit: $$\gamma_\mathcal{T}=\lim_{n\to\infty}\left(-\frac{i}{2}\sum_{k=1}^n ...
3
votes
2answers
331 views

Is there any numerical technique to sum x^(n^alpha), n=0,1,…?

I've come across this infinite series: $\sum_{n=0}^\infty x^{n^\alpha}$, with $0<x<1$ and $\alpha > 0$. Does this series have a name and/or is there a method for computing it (besides brute ...
2
votes
1answer
257 views

Can we find an l-2 sequence if we know all l-p norms?

I'm wondering if there is a way to approximate the first $M$ terms of a non-increasing $\ell^2$ sequence $\{c_n\}$ if we know $|c|_p^p = \sum c_n^p$ for $p=2,3,4,\dots$? I've tried truncating the ...