<s>[INST] Why does $\sum_{n=1}^\infty \sqrt{n+1}-\sqrt{n}$ diverge? Why does $\sum_{n=1}^\infty \sqrt{n+1}-\sqrt{n}$ diverge?
Using the ratio test I get the following. First of all since $u_n=\sqrt{n+1}-\sqrt{n}=(\sqrt{n+1}-\sqrt{n})\frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+1}+\sqrt{n}}=\frac{1}{\sqrt{n+1}+\sqrt{n}}$
Then $\left| \frac{u_{n+1}}{u_n} \right|=\frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+2}+\sqrt{n+1}}\lt 1$.
By the ratio test, the series should converge, but it does not. What am I doing wrong?
 [/INST] The ratio test actually doesn't conclude anything. Note that
$$
\frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+2}+\sqrt{n+1}} = \frac{\sqrt{1 + 1/n} + 1}{\sqrt{1+2/n}+\sqrt{1+1/n}} \to \frac{1+1}{1+1} = 1
$$
Now notice that for large $n$ we have
$$
n > 2\sqrt{n+1} > \sqrt{n+1} + \sqrt{n} \implies \frac{1}{n} < \frac{1}{\sqrt{n+1}+\sqrt{n}}
$$
So we get that the series diverges.
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