Convergence of $\sum_{n=1}^{\infty} \sqrt{a_n a_{n+1}}$ and $\sum_{n=1}^{\infty} \sqrt{a_n}/n^p$ for $p>1/2$ and $\sum_{n=1}^{\infty} a_n$ convergent

Question

Let $\langle a_n \rangle$ be a sequence of positive numbers. If $p>1/2$ and $\sum_{n=1}^{\infty} a_n$ converges, then how can I prove that $\sum_{n=1}^{\infty} \sqrt{a_n a_{n+1}}$ and $\sum_{n=1}^{\infty} \sqrt{a_n}/n^p$ also converges? It seems that any kind of convergence test does not work.

Also, does the converse hold? I think it doesn't, but I have trouble finding a counterexample.

Answer 1

Hints: (substantial)

• For the first, use the AM–GM inequality: $$ \sqrt{a_na_{n+1}} \leq \frac{a_n+a_{n+1}}{2} $$ and comparison theorems.

• For the second, use the Cauchy—Schwarz inequality* on the partial sums: $$ \sum_{n=1}^N \frac{\sqrt{a_n}}{n^p} \leq \sqrt{\sum_{n=1}^N a_n}\sqrt{\sum_{n=1}^N \frac{1}{n^{2p}}} $$ then the fact that $p>\frac{1}{2}$ and and comparison theorems.

For the counterexamples, you may want to find sequences such that

• first: $a_{2n}$ is always "big," yet $a_{2n+1}$ is really "small."
• second: for $p=1$, $\frac{\sqrt{a_n}}{n} = \frac{1}{n \ln^2 n}$ (recall that $\sum_{n=1}^\infty \frac{1}{n \ln^2 n}$ is a convergent series)

$\ast$ As mentioned in a comment below by Martin R, the AM–GM inequality would do as well for the second (and Cauchy–Schwarz for the first), but it's good to know several tools.