<s>[INST] $\sum_k\left(\frac{a_k}{c_k}\right )^{2}$, $a_k=\frac{1}{k}$, $c_k$=$1$, $\frac{1}{2}$,$\frac{1}{2}$,$\frac{1}{3}$,$\frac{1}{3}$,$\frac{1}{3}$, ... With $a_k$ and $c_k$ (1 repeating once, then 1/2 twice, then 1/3 three times, then 1/4 4 times, etc) as defined above, does the sum converge?
 [/INST] That's a bit awkwardly defined, I'd simply write
$$S = \sum_{k=1}^\infty \left(\frac{c_k}{k} \right )^{2}$$
with $c_k = 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots$.
We have $c_k =\lfloor \sqrt{2k} + \frac{1}{2}\rfloor$.
But we have $\lfloor \sqrt{2k} + \frac{1}{2}\rfloor \geq \sqrt{k}$ for sufficiently large $k$, yet
$$ \sum_{k=1}^\infty \left(\frac{\sqrt{k}}{k} \right )^{2} = \sum_{k=1}^\infty \frac{1}{k}$$
diverges, so your sum must as well.
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