.......(i) Let n4+n2+1=[(n2)2+12+2(n2)(1)]−n2 =(n2+1)2−n2 =(n2+n+1)(n2−n+1) .........(ii) Let un=tan−1(
2n
n4+n2+2
) =tan−1
2n
1+(n4+n2+1)
=tan−1
(n2+n+1)−(n2−n+1)
1+(n2+n+1)(n2−n+1)
u1=tan−1(n2+n+1)−tan−1(n2−n+1) On putting n = 1, 2 , 3 ....... successively in Eq. (iii), we get u1=tan−13−tan−11 u2=tan−17−tan−13 u3=tan−113−tan−17 ................................. un=tan−1(n2+n+1)−tan−1(n2−n+1) On adding vertically, we get