I'm trying to figure out the limit of the sequence
\begin{equation*} \lbrace\sum_{n=1}^{k}\frac{1}{\sqrt{k^2 + n}}\rbrace_{k=1}^\infty \end{equation*}
I feel like I'm close to understanding it, I just need a little push. I understand that I need to show that this sequence lies between $\lbrace \frac{k}{\sqrt{k^2 + k}}\rbrace_{k=1}^\infty$ and $\lbrace \frac{k}{\sqrt{k^2 + 1}}\rbrace_{k=1}^\infty$, and that both of those sequences converge to 1. However, I'm not quite sure how to show that both of them converge to 1, or why the first sequence would lie between them. The summation and the square roots are confusing me, I think.
Edit: Okay, so I have the notion that I should divide everything in the convergent sequences by k, but I don't think I'm doing the algebra right. For the first sequence, dividing by k seems to get me $\frac{1}{\sqrt{k + 1}}$, which seems to converge to 0 as k approaches $\infty$ since $\frac{1}{\sqrt{k + 1}} \leq \frac{1}{k}$. Similarly, dividing everything in the latter sequence seems to give $\frac{1}{\sqrt{k + \frac{1}{k}}}$, which would also converge to zero as k approaches $\infty$. I don't feel like I'm distributing the k properly over the square root.