Consider SN=sum(n=1 to N) (1/(n^2+2n))

Write a formula for SN and find lim n-->infinity SN

So the terms are like 1/3+1/8+1/15+1/24.... essentially we are adding 5,7,9,11...etc..the odd numbers starting from 5 to the bottom. I don't know how to write a formula for this that is not recursive though.

I got SN=(SN-1)/(2N+1), but I think they want a formula that doesn't use recursion.