The sum of the the series cot−1⁡2+cot−1⁡8+cot−1⁡18+cot−1⁡32+…….. is

The given series=cot−1⁡2(1)2+cot2⁡2(2)2+cot−1⁡2(3)2+cot−1⁡2(4)2+…∴Tn=cot−1⁡2n2=tan−1⁡12n2=tan−1⁡(2n+1)−(2n−1)1+(2n+1)(2n−1)=tan−1⁡(2n+1)−tan−1⁡(2n−1)∴T1=tan−1⁡3−tan−1⁡1, T2=tan−1⁡5−tan−1⁡3,T3=tan−1⁡7−tan−1⁡5and so on.∴Sn=tan−1⁡(2n+1)−tan−1⁡1∴limn→∞ Sn=tan−1⁡∞−π4=π2−π4=π4