The sum of the the series cot−12+cot−18+cot−118+cot−132+…….. is
π2
π4
π6
none of these
The given series
=cot−12(1)2+cot22(2)2+cot−12(3)2+cot−12(4)2+…
∴Tn=cot−12n2=tan−112n2=tan−1(2n+1)−(2n−1)1+(2n+1)(2n−1)=tan−1(2n+1)−tan−1(2n−1)∴T1=tan−13−tan−11, T2=tan−15−tan−13,T3=tan−17−tan−15
and so on.
∴Sn=tan−1(2n+1)−tan−11∴limn→∞ Sn=tan−1∞−π4=π2−π4=π4