- #1
arpon
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Homework Statement
$$ \sum_{n=1}^\infty\frac{1}{1+(a+nb)^2} = ? $$
2. The attempt at a solution
I approximated the result by integration,
$$
\begin{align}
\sum_{n=1}^\infty \frac{1}{1+(a+nb)^2} &\approx \lim_{N \rightarrow +\infty} {\int_{0}^N \frac{1}{1+(a+bx)^2} dx}\\
&= \lim_{N \rightarrow +\infty} {\frac{tan^{-1} (a + Nb)}{b}} - \frac{tan^{-1} (a)}{b}\\
&= \frac{\pi}{2b} - \frac{tan^{-1} (a)}{b}
\end{align}
$$
Using this integration method, I further proved,
$$\frac{\pi}{2b} - \frac{tan^{-1} (a+b)}{b}< \sum_{n=1}^\infty \frac{1}{1+(a+nb)^2} < \frac{\pi}{2b} - \frac{tan^{-1} (a)}{b}$$
But, how can I calculate the exact result?
I faced this problem when solving a physics problem.
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