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Kalman - Six ways to sum a series discusses Euler's original proof for the Basel problem $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $: $$\frac{\sin(\sqrt x)}{\sqrt x} = 1- \frac{x}{3!}+ \...

This question is inspired by the MO query here, although it has no direct implications. Define the family of polynomial functions $$f_n(x)=n^2x^{n-1}-\frac{d}{dx}\left(\frac{x^n-1}{x-1}\right),$$ and ...

Hilbert's seventeenth problem asks to prove that every positive semidefinite form can be written as the sum of squares of rational functions. Currently we don't seem to have a good understanding of ...