<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] Another possible solution:
$$
0 < a_n := \frac{1}{\sqrt{4n + 1}} - \frac{1}{\sqrt{4n + 3}}
 = \frac{1}{\sqrt{4n + 1}} - \frac{1}{\sqrt{4(n+1) - 1}} \\
 < \frac{1}{\sqrt{4n + 1}} - \frac{1}{\sqrt{4(n+1) + 1}} =: b_n
$$
and $\sum b_n$ is convergent as a telescoping series.
It follows that $\sum a_n$ is convergent and
$$
 \sum_{n=0}^\infty a_n < \sum_{n=0}^\infty b_n = 1 \quad .
$$
Or: The terms are positive, and the partial sums
$$
\sum_{n=0}^N \frac{1}{\sqrt{4n + 1}} - \frac{1}{\sqrt{4n + 3}} =
1 + \sum_{n=1}^N \left(-\frac{1}{\sqrt{4n - 1}} + \frac{1}{\sqrt{4n + 1}} \right) - \frac{1}{\sqrt{4N + 3}} < 1
$$
are bounded. (This is essentially what achille hui suggested
in the above comment.)
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