If the sum of $f(n)$ diverge, then does the sum of $\sqrt{f(n)}$ diverge? Lets say that $$\sum_{n=a}^\infty f(n)$$ diverges. Does $$\sum_{n=a}^\infty \sqrt{f(n)}$$ necessarily diverge?

Suppose $\sum f(n)$ diverges. If $f(n)$ is unbounded or does not vanish, the result is immediate, so assume it is bounded and vanishing.

Observe that there exists some $k>0$ such that $f(n)<1$ for all $n>k$. This implies that $\sqrt{f(n)}>f(n)$ for any $n>k$. Since we know $\sum f(n)$ diverges, so must this series whose tail is composed of larger terms.

EDIT NOTE: This argument assumes that all terms stay in $\mathbb{R}$