Limits of Series

[Ed Quanta]

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?

[cogito²]

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?
Do you mean the series:

\lim_{n \to \infty} \sum_{i=1}^n \frac{1}{\sqrt{n^2 + n}} \ \ ?

Because that series does not converge.

[Ed Quanta]

Doesn't it converge to 1?

[matt grime]

Nope, since

n^2+n < n^2+2n+1

the sum you've got there, modulo some initial terms, is greater than

\sum \frac{1}{n+1}

so it diverges.

NB. cogito's post uses n twice as the parameter and the limit, it should be i inside the sum.