Convergence of $\sum\sqrt{a_n}$ and $\sum\frac{a_1+\ldots+a_n}{n}$ when $\sum_{n=1}a_n,\ a_n>0$ is a convergent series Let $\sum_{n=1}a_n,\ a_n>0$ be a convergent series. Consider following statements:


*

*P: The series $\sum_{n=1}\sqrt{a_n}$ is always convergent

*Q: The series $\sum_{n=1} \frac{a_1 + a_2 +...+a_n}{n}$ is always divergent


Then


*

*Both P and Q are true,

*P is true but Q is false,

*Both P and Q are false,

*P is false and Q is true


I was thinking of suitable examples/counterexamples but I could not find any. Please suggest.
Thanks

*

*For P. Let $a_n=\frac{1}{n^2}$ as you guessed. Then $\sum_{n=1}\frac1{n^2}$ converges, while $$\sum_{n=1}\sqrt{\frac1{n^2}}=\sum_{n=1}\frac1{n}$$ does not.

*For Q. Since $a_n>0$ for all $n\in \mathbb N$, you have that $$\sum_{n=1}\frac{a_1+\ldots+a_n}{n}>\sum_{n=1}\frac{a_1}{n}=a_1\sum_{n=1}\frac1n$$ which diverges. So, by comparison the LHS always diverges.


In sum, P is false, Q is true.
