Question

Given that ∑(n=1 to inf) 1/(n^2) = (pi^2)/6, find the value of ∑(n=1 to inf) ((5n^2+6n+3)/((n^2)((1+n)^2))).

Answer 1

I think it should help to notice that

5n^2 + 6n + 3 = 3n^2 + 6n + 3 + 2n^2 = 3(n+1)^2 + 2n^2

So, what you have is just

3/n^2 + 2/(n+1)^2