<s>[INST] Is $\lim_{n\to\infty} \left( (\sum^n_{m=1} \frac{1}{\sqrt{m}}) - \sqrt{n} \right)$ convergent? 
$\lim_{n\to\infty} \left( (\sum^n_{m=1} \frac{1}{\sqrt{m}}) - \sqrt{n} \right)$ : convergent?

My attempt
$$
(\sum^n_{m=1} \frac{1}{\sqrt{m}}) - \sqrt{n} = (\frac{1}{\sqrt{1}} - \frac{1}{\sqrt{n}}) + (\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{n}}) + \cdots + (\frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n}})
$$
But I stuck here.
 [/INST] The limit is not convergent.  You can use the integral test to see this.
\begin{align}
\sum_{m=1} ^n  \frac{1}{\sqrt{m}} & \geq \int_{1}^{n} \frac{1}{\sqrt{x}} \ dx \\ & = 2\sqrt{n} - 2 .
\end{align}
So we can conclude that
$$\sum_{m=1} ^n  \frac{1}{\sqrt{m}} - \sqrt{n} \ \geq \sqrt{n}- 2.$$
Letting $n \to \infty$ givs us what we want.
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