<s>[INST] Convergent series after rearrangement becomes divergent Show that $$1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}}+\frac{1}{\sqrt{5}}-\cdots$$
is convergent but by rearrangement the following series
 $$\left(1+\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{2}}\right)+\left(\frac{1}{\sqrt{5}}+\frac{1}{\sqrt{7}}-\frac{1}{\sqrt{4}}\right)+\cdots $$
 is divergent.
Attempt:

The 1st series $$\sum_{n=1}^\infty (-1)^{n+1} \frac{1}{\sqrt{n}}$$ can be proved to be convergent by Leibnitz Test as
(1) $u_n= \frac{1}{\sqrt{n}}\to 0$ as $n\to\infty$ 
(2) $\{u_n\}$ is monotone decreasing

But please help me to show that the second series, after rearrangement, is divergent.
 [/INST] Hint: for $n\ge 0:$
$$\dfrac{1}{\sqrt{4n+1}} + \dfrac{1}{\sqrt{4n+3}} - \dfrac{1}{\sqrt{2(n+1)}}>\dfrac{1}{\sqrt{4n+4}} + \dfrac{1}{\sqrt{4n+4}} - \dfrac{1}{\sqrt{2(n+1)}}=\\
\left(1-\frac{1}{\sqrt{2}}\right)\cdot \frac{1}{\sqrt{n+1}}$$
</s>