S_n = sum_(k=1)^n (n+1)/(a n^2+(n(n+1))/2)(k/(n+1))^aSn=n∑k=1n+1an2+n(n+1)2(kn+1)a or
S_n = (2(n+1))/(2an+n+1)sum_(k=1)^n (k/(n+1))^a1/nSn=2(n+1)2an+n+1n∑k=1(kn+1)a1n or
S_n = 2/(2an/(n+1)+1)sum_(k=1)^n (k/(n+1))^a1/nSn=22ann+1+1n∑k=1(kn+1)a1n then
lim_(n->oo)S_n =2/(2a+1) int_0^1xi^a d xi = 2/(2a+1)1/(a+1)
Now 2/((2a+1)(a+1))=1/60 and solving for a
a = {-17/2, 7} rArr a = 7