Q:

If $\sum a_n$ with $a_n>0$ is convergent, then is $\sum {a_n}^2$ and $\sum \sqrt{a_n}$ always convergent?

If $\sum a_n$ with $a_n>0$ is convergent, then is $\sum {a_n}^2$ and $\sum \sqrt{a_n}$ and $\sum \sqrt{a_na_{n+1}}$ always  convergent? Either prove it or give a counter example.
 I am thinking that the second series is convergent by comparision test. because $$ \sqrt{a_n}\leq a_n$$

A:

If $\sum\limits_n a_n$ is convergent, then $a_n <1$ for every $n >N$, hence $\sum\limits_na^2_n$ converges too. At the same time $\sum\limits_n\sqrt{a_n}$ need not be necessarily convergent. Take e.g. $a_n= \frac{1}{n^2}$.
EDIT thi is of course based on the fact that for $0<x<1$ of course $0 < x^2 < x <\sqrt{x}<1$.

