<s>[INST] Convergence of $\sum \frac{1}{\sqrt{4n + 1}} - \frac{1}{\sqrt{4n + 3}}$ I need to prove $\sum \frac{1}{\sqrt{4n + 1}} - \frac{1}{\sqrt{4n + 3}}$ converges. The root test is inconclusive, so I check in W.A., $\sum \frac{1}{\sqrt{4n + 1}} - \frac{1}{\sqrt{4n + 3}}$ converges by comparison test, but I don't know what series to compare. I've tried to compare with $\sum \frac{1}{4n+1} - \frac{1}{4n + 3}$ because this serie converges to $\frac{\pi}{4}$, but we have exactly the opposite inaquality $\frac{1}{\sqrt{4n + 1}} - \frac{1}{\sqrt{4n + 3}} > \frac{1}{4n + 1} - \frac{1}{4n + 3}$.
Can you help me?
 [/INST] Asmptotics.
As $n \to +\infty$:
$$\begin{align}
(4n+1)^{-1/2} &= (4n)^{-1/2}\left(1+\frac{1}{4n}\right)^{-1/2}
=\frac{1}{2\sqrt{n}}\left(1-\frac{1}{8n}+O(n^{-2})\right)
\\
(4n+3)^{-1/2} &= (4n)^{-1/2}\left(1+\frac{3}{4n}\right)^{-1/2}
=\frac{1}{2\sqrt{n}}\left(1-\frac{3}{8n}+O(n^{-2})\right)
\\
(4n+1)^{-1/2} - (4n+3)^{-1/2}&=
\frac{1}{2\sqrt{n}}\left(0+\frac{2}{8n}+O(n^{-2})\right)
=\frac{1}{8n^{3/2}}+O(n^{-5/2})
\end{align}$$
The series converges by comparison with $$\sum\frac{1}{8n^{3/2}}$$
</s>