Infinte Series with Conditions I was wondering if someone could give me a hint about how to get started on this problem, and/or hide the solution to it. I really want to try it on my own first. 
Suppose $a_k \geq 0$ for all $k \in \mathbb{N}$ and $\sum a_k < \infty$. Either prove that the given series converges, or give an example for which it diverges:
$\sum_{k=1}^{\infty} \sqrt{\frac{a_k}k}$.

HINT $1$: Look for a counterexample. 
If $b_k=\sqrt{a_k/k}$, you want $\sum_kb_k$ to diverge while $\sum_ka_k=\sum_kkb_k^2$ converges, so $\langle kb_k:k\ge 1\rangle$ has to converge to $0$ faster than $\langle b_k:k\ge 1\rangle$. It’s a good bet that $\sum_ka_k$ will be just barely convergent, while $\sum_kb_k$ will just barely diverge. The most obvious ‘barely divergent’ series is the harmonic series, but it’s not good enough: you’ll need $\sum_kb_k$ to diverge even slower than the harmonic series.
HINT $2$:

 Try $b_k=\dfrac1{kf(k)}$ for some slowly growing function $f$, so that $a_k=\dfrac1{k(f(k))^2}$.

HINT $3$:

 What is $\displaystyle\int_2^\infty\frac{dx}{x(\ln x)^2}$? Does this give you any ideas for $a_k$?

There may well be other nice solutions; this was the first that occurred to me.
