Summation of the Inverted Tangents

Geometry Level 3

Evaluate $\large \sum_{m=1}^{\infty} \tan^{-1} \left(\frac{8m}{16m^4-32m^2+5}\right).$

Remember that the range of the inverse tangent function is $( - \frac{ \pi}{2} , \frac{ \pi }{2})$ .

\tan^{-1}(2)-\frac{\pi}{2}

\frac{\pi}{4}+\tan^{-1}(3)

\frac{\pi}{2} - \tan^{-1}(2)

\frac{\pi}{2}+\tan^{-1}(2)