$\sum_k\left(\frac{a_k}{c_k}\right )^{2}$, $a_k=\frac{1}{k}$, $c_k$=$1$, $\frac{1}{2}$,$\frac{1}{2}$,$\frac{1}{3}$,$\frac{1}{3}$,$\frac{1}{3}$, ...

Question

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?

Answer 1

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.