Does the sequence converge or diverge?

Question

I'm having trouble understanding how to do this one. If anyone could help I would be grateful. Does the sequence $$ \left\{ \sum_{n=1}^k \left( \frac{1}{\sqrt{k^2+n}} \right) \right\}_{k=1}^\infty $$ diverge or converge? If the sequence converges, find the limit.

Answer 1

Hint: For large $k$, $$ \sum_{n=1}^k \frac{1}{\sqrt{k^2+n}} \approx \sum_{n=1}^k \frac{1}{\sqrt{k^2}} = \sum_{n=1}^k \frac{1}{k} = 1. $$