[Math] If $a_n\ge0$ and $\sum a_n$ converges then $\sum\sqrt{a_na_{n-1}}$ converges, what about the converse

Question

1. Suppose the series $\sum_{n=1}^{\infty}{a_n}$ is convergent ($a_n \geq0$), Is it true that $\sum_{n=1}^{\infty}\sqrt{a_na_{n-1}}$ is convergent ?

2. Is the converse true?

My attempt:

The first part I was able to do using AM-GM inequality and comparison test for series.

But I am not able to prove or generate a counter example for the second part.

Answer 1

No, it is not true. Define $a_n = n^2$ for $n$ even and $a_n = n^{-100}$ for $n$ odd. Then clearly $\sum a_n$ is divergent, but each term

$$\sqrt{a_n a_{n - 1}} \sim n^{-49}$$

gives a convergent series.