AoPS

MATHEMATICS · Single correct · AoPS · Mathematics · Sequences & Series · Sequences & Series

Problem

∑r=1∞tan⁡−1(2(r+2)2)\displaystyle\sum_{r=1}^{\infty}\tan^{-1}\left(\dfrac{2}{(r+2)^2}\right) is (1) π−(tan⁡−12+tan⁡−13)\pi-\left(\tan^{-1}2+\tan^{-1}3\right) (2) π4\dfrac{\pi}{4} (3) 7π4\dfrac{7\pi}{4} (4) π+(tan⁡−12+tan⁡−13)\pi+\left(\tan^{-1}2+\tan^{-1}3\right)

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