I'm trying to find the value of the following sum (if exist):
$$\sum_{n=1}^{\infty}\left(\arctan\left(\frac{1}{4}-n\right)-\arctan\left(-\frac{1}{4}-n\right)\right)$$
where, $\arctan$ represent the inverse tangent function – $\tan^{-1}$.
I tried to use the telescoping series idea and the sequence of partial sums but I couldn't cancel any terms!
Best Answer
Consider $$\tag{1}f(x):=\sum_{n=1}^\infty \arctan\left(\left(\frac 14-n\right)x\right)-\arctan\left(\left(-\frac 14-n\right)x\right)$$ and let's rewrite the derivative of $f$ : \begin{align} \tag{2}f'(x)&=\frac 4{x^2}\sum_{n=1}^\infty\frac{1-4n}{(4n-1)^2+\bigl(\frac 4x\bigr)^2}-\frac{-1-4n}{(4n+1)^2+\bigl(\frac 4x\bigr)^2}\\ \tag{3}f'(x)&=\frac 4{x^2}\left(\frac {-1}{1^2+\left(\frac 4x\right)^2}+\sum_{k=1}^\infty\frac{k\sin\bigl(k\frac {\pi}2\bigr)}{k^2+\bigl(\frac 4x\bigr)^2}\right)\\ \end{align} (since the $k=1$ term didn't appear in $(2)$)
But the series in $(3)$ may be obtained from $\,\frac d{d\theta} C_a(\theta)\,$ with : $$\tag{4}C_a(\theta)=\frac {\pi}{2a}\frac{\cosh((\pi-|\theta|)a)}{\sinh(\pi a)}-\frac 1{2a^2}=\sum_{k=1}^\infty\frac{\cos(k\,\theta)}{k^2+a^2}$$ which may be obtained from the $\cos(zx)$ formula here (with substitutions $\ x\to\pi-\theta,\ z\to ia$).
The replacement of the series in $(3)$ by $\,\frac d{d\theta} C_a(\theta)\,$ applied at $\,\theta=\frac {\pi}2$ gives us : \begin{align} f'(x)&=\left(-\arctan\left(\frac x4\right)\right)'-\frac 4{x^2}C_{\frac 4x}\left(\theta\right)'_{\theta=\frac {\pi}2}\\ &=\left(-\arctan\left(\frac x4\right)\right)'+\frac{4\pi}{2x^2}\frac{\sinh\left(\frac{\pi}2 \frac 4x\right)}{\sinh\left(\pi\frac 4x\right)}\\ &=\left(-\arctan\left(\frac x4\right)\right)'+\frac{\pi}{x^2}\frac 1{\cosh\left(\frac{2\pi}x\right)}\\ \end{align}
Integrating both terms returns (with constant of integration $\frac {\pi}2$ since $f(0)=0$) : $$f(x)=\frac {\pi}2-\arctan\left(\frac x4\right)-\arctan\left(\tanh\left(\frac {\pi}x\right)\right)\quad\text{for}\ \ x>0$$ i.e. the neat : $$\tag{5}\boxed{\displaystyle f(x)=\arctan\left(\frac 4x\right)-\arctan\left(\tanh\left(\frac {\pi}x\right)\right)}\quad\text{for}\ \ x>0$$
So that your solution will be (for $x=1$) : $$\boxed{\displaystyle \arctan(4)-\arctan\left(\tanh(\pi)\right)}$$