Sum of infinite terms of the series $$cot$$−$$1$$⁡$$12+34+$$$$cot$$−$$1$$⁡$$22+34+$$$$cot$$−$$1$$⁡$$32+34+$$… is

Question

Sum of infinite terms of the series $$cot$$−$$1$$⁡$$12+34+$$$$cot$$−$$1$$⁡$$22+34+$$$$cot$$−$$1$$⁡$$32+34+$$… is

Infinity Learn · Accepted Answer

Let $$Tn=$$$$cot$$−$$1$$⁡$$n2+34$$⇒$$Tn=$$$$cot$$−$$1$$⁡$$4n2+34$$⇒$$Tn=$$$$tan$$−$$1$$⁡$$44n2+3=$$$$tan$$−$$1$$⁡$$1n2+34$$⇒$$Tn=$$$$tan$$−$$1$$⁡$$11+n2$$−$$14$$⇒$$Tn=$$$$tan$$−$$1$$⁡$$11+n+12n$$−$$12$$⇒$$Tn=$$$$tan$$−$$1$$⁡$$n+12$$−n−$$121+n+12n$$−$$12$$⇒$$Tn=$$$$tan$$−$$1$$⁡$$n+12$$−$$tan$$−$$1$$⁡n−$$12Now$$ $$Sn=T1+T2+T3+$$…$$+Tn$$⇒$$Sn=$$$$tan$$−$$1$$⁡$$32$$−$$tan$$−$$1$$⁡$$12+$$$$tan$$−$$1$$⁡$$52$$−$$tan$$−$$1$$⁡$$32+$$$$tan$$−$$1$$⁡$$72$$−$$tan$$−$$1$$⁡$$52+$$…$$+$$$$tan$$−$$1$$⁡$$n+12$$−$$tan$$−$$1$$⁡$$n+12$$∴S∞$$=$$$$tan$$−$$1$$⁡∞−$$tan$$−$$1$$⁡$$12=$$π$$2$$−$$tan$$−$$1$$⁡$$12=$$$$cot$$−$$1$$⁡$$12$$∴S∞$$=$$$$tan$$−$$1$$⁡$$2$$

Sum of infinite terms of the series $\cot^{- 1} (1^{2} + \frac{3}{4}) + \cot^{- 1} (2^{2} + \frac{3}{4}) + \cot^{- 1} (3^{2} + \frac{3}{4}) + \dots$ is

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Sum of infinite terms of the series cot−1⁡12+34+cot−1⁡22+34+cot−1⁡32+34+… is

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Sum of infinite terms of the series $\cot^{- 1} (1^{2} + \frac{3}{4}) + \cot^{- 1} (2^{2} + \frac{3}{4}) + \cot^{- 1} (3^{2} + \frac{3}{4}) + \dots$ is