Analytical solution of a convergent series

Question

Which is the analytical solution of the infinite sum $\frac{1}{1+1^{2}}+\frac{1}{1+2^{2}}+\frac{1}{1+3^{2}}+\frac{1}{1+4^{2}}+\cdots$?

Answer 1

We start with this identity for the sine function: $$ \frac{\sinh x}{x} = \prod_{k=1}^\infty \left(1+\frac{x^2}{k^2\pi^2}\right) $$

For details on its derivation, check this article. This product formula was already used to prove another interesting identity involving $\pi$ and infinite series!

Well, we also know that $$ \frac{d}{dx} \log \sinh x = \coth x. $$

Due to the properties of the logarithm function, $\log \sinh x$ is $$ \log x + \sum_{k=1}^\infty \log \left(1+\frac{x^2}{k^2\pi^2}\right), $$ therefore, $$ \coth x = \frac{1}{x} + 2x \sum_{k=1}^\infty \frac{1}{k^2 \pi^2 + x^2}. $$

Using $x = \pi$ leads to $$ \coth \pi = \frac{1}{\pi} + \frac{2}{\pi} \sum_{k=1}^\infty \frac{1}{k^2 + 1}, $$ then, solving for the required summation, $$ \sum_{k=1}^\infty \frac{1}{k^2 + 1} = \frac{\pi \coth \pi - 1}{2}. $$