# What is the value of the series $\sum_{n\geq 1}\frac{1}{n^2+1}$?

What is the value of $$\sum_{n=1}^{\infty}\frac1{1+n^2}$$ I don't know how to solve it.
I tried to solve it by power series, but I failed.

• I'm afraid it's one of those "inspection" answers, where you have to draw upon your knowledge of familiar and well known series formulas -- probably a disappointing solution! – Hugh Entwistle Oct 15 '16 at 11:07

## 5 Answers

Hint: $$\frac{\pi x}{2}\coth \pi x=\frac{1}{2}+\sum_{n=1}^{\infty }\frac{1}{\left ( \frac{n}{x} \right )^2+1}$$ at $x=1$ $$\sum_{n=1}^{\infty }\frac{1}{n^2+1}=\frac{\pi}{2}\coth \pi -\frac{1}{2}$$

I will give an elementary solution based on the following lemmas:

Lemma 1. For any $n\in\mathbb{N}^+$, $$\int_{0}^{+\infty}\frac{\sin(nx)}{n}e^{-x}\,dx = \frac{1}{1+n^2}.$$ Proof: integration by parts.

Lemma 2. The following series is the Fourier series of a $2\pi$-periodic sawtooth wave $f(x)$,
that over the interval $(0,2\pi)$ equals $\frac{\pi-x}{2}$: $$f(x)=\sum_{n\geq 1}\frac{\sin nx}{n}$$ Proof: direct computation of Fourier coefficients.

By Lemma 1 and Lemma 2, $$\begin{eqnarray*}\sum_{n\geq 1}\frac{1}{n^2+1}&=&\int_{0}^{+\infty}f(x)e^{-x}\,dx\\(\text{periodicity of } f(x))\qquad&=&\left(1+\frac{1}{e^{2\pi}}+\frac{1}{e^{4\pi}}+\ldots\right)\int_{0}^{2\pi}f(x)e^{-x}\,dx\\[0.2cm]&=&\frac{e^{2\pi}}{e^{2\pi}-1}\int_{0}^{2\pi}\frac{\pi-x}{2}e^{-x}\,dx\\[0.2cm](\text{direct computation})\qquad &=&\color{red}{\frac{\pi\coth\pi-1}{2}}.\end{eqnarray*}$$

Hint. It's well-known that $$\pi\cot(\pi z)=\frac{1}{z}+2\sum_{n=1}^\infty \frac{z}{z^2-n^2}$$


• This is exactly how I would do it :) – Arturo don Juan Oct 18 '16 at 0:56
• @ArturodonJuan Thanks. It's a straightforward method and it's widely used ( and explained ) in the old Abramowitz and Stegun Table. – Felix Marin Oct 18 '16 at 0:58
• Yeah I know, I'm a physics student who learned it from Arfken's book (and they cite Abramowitz and Stegun) – Arturo don Juan Oct 18 '16 at 1:03

It is shown in this answer that $$\mathrm{PV}\sum_{n\in\mathbb{Z}}\frac1{n+z}=\pi\cot(\pi z)\tag{1}$$ converges for all $z\in\mathbb{C}$. Therefore, \begin{align} \sum_{n=1}^\infty\frac1{n^2+z^2} &=\frac i{2z}\sum_{n=1}^\infty\left(\frac1{n+iz}+\frac1{-n+iz}\right)\\ &=\frac i{2z}\left[\mathrm{PV}\sum_{n\in\mathbb{Z}}\frac1{n+iz}-\frac1{iz}\right]\\ &=-\frac1{2z^2}+\frac{\pi i}{2z}\cot(\pi iz)\\[6pt] &=-\frac1{2z^2}+\frac\pi{2z}\coth(\pi z)\tag{2} \end{align} Plugging in $z=1$ gives $$\sum_{n=1}^\infty\frac1{n^2+1}=-\frac12+\frac\pi2\coth(\pi)\tag{3}$$