Convergence of unknown series

Question

I have the following problem and I don't actually know how to solve it.

Let $x_n$, n belongs to natural numbers set such that ∑ $x_n$, n≥1 is convergent. Prove that the series $\sum \sqrt{x_nx_{n+1}}$ is convergent.

Can someone please just give me a hint on how should I resolve this?

Answer 1

For $x_{n}\geq 0$, \begin{align*} \sum\sqrt{x_{n}x_{n+1}}\leq\left(\sum x_{n}\right)^{1/2}\left(\sum x_{n+1}\right)^{1/2}<\infty \end{align*} by Cauchy-Schwarz inequality.

Answer 2

Hint Since $2ab \leq a^2+b^2$ you have $$\sqrt{x_nx_{n+1}}\leq \frac{1}{2} x_n+ \frac{1}{2}x_{n+1}$$ and $\sum_n x_n$ and $\sum_n x_{n+1}$ are convergent.