Prove $\sinh1\Bigl(1+2{\sum}_{n=1}^{\infty}\frac{(-1)^{n}}{(n\pi)^2+1}\Bigr)=1$ for $-1\lt x \lt 1$
Given a Fourier series $f(x)$: What's the difference between the value the expansion takes for given $x$ and the value it converges to for given $x$?