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

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=π2−tan−1⁡12=cot−1⁡12∴S∞=tan−1⁡2