How do you prove the limit of a series? For example, bn=1/square root of (n^2+1)+ 1/square root of (n^2+2) + ...+1/square root of (n^2 + n). I know this series converges since the summation of 1/n^2 does. But the only reason this is because of integration. How could I know that some series converges without using integrals, and then be able to find to what it converges?