Is there a converging sum of the form $0<\lim_{n \to \infty }\sum_n^{n^2} a_k <\infty\, , a_k>0$ looking at this question Can we find the limit of the sequence $a_n=\frac{1}{n^2}+\frac{1}{(n+1)^2}+\frac{1}{(n+2)^2}+\dots+\frac{1}{(2n)^2}$ using Riemann sum?
Makes one wonder if there is a converging sum s.t. $0<\lim_{n \to \infty} \sum_n^{n^2} a_k <\infty , a_k>0$.
Tried using $a_k=\frac{1}{k}+\frac{1}{k+1}+\frac{1}{k+2}+\dots+\frac{1}{2k}$ so that maybe end up with some result looking like $(\ln 2)^2$ or $\ln 4$ or anything related to some $g(\ln 2)$ where $g$ is some function.
Of course if it is proved that if $\lim_{n \to \infty} \sum_n^{n^2} a_n <\infty , a_n>0$ is always (div/con)verges then $\lim_{n \to \infty} \sum_n^{n^\gamma} a_n <\infty , a_n>0$ would also (div/con)verge.
 A: Let $a_k = 1/(k\ln k)$ for $k>1.$ Set $S_n = \sum_{k=n}^{n^2}a_k.$ Then for $n>1,$
$$\int_n^{n^2}\frac{1}{x\ln x}\,dx < S_n < \frac{1}{n\ln n} + \int_n^{n^2}\frac{1}{x\ln x}\,dx.$$
That integral miraculously equals $\ln 2$ for all $n>1.$ Therefore $\lim_{n\to \infty} S_n = \ln 2.$
A: This is not a rigorous answer.
$$a_k=\psi ^{(1)}(k)-\psi ^{(1)}(2 k+1)$$ Considering large values of $k$ and the asymptotics of the polygamma function, then $$a_k=\frac{1}{2 k}+\frac{5}{8 k^2}+\frac{7}{48 k^3}-\frac{31}{960
   k^5}+O\left(\frac{1}{k^6}\right)$$ Summing again from $n$ to $n^2$ and expanding again the resulting polygamma functions, we end with 
$$\sum_n^{n^2} a_k=\frac{1}{2} \log \left({n}\right)+\frac{7}{8 n}+\frac{5}{96
   n^2}+\frac{17}{96 n^3}+O\left(\frac{1}{n^4}\right)$$ Just for illustration purposes, a few numerical values 
$$\left(
\begin{array}{cccc}
 n & \text{exact} & \text{approximation} & \Delta \\
 5  & 0.98356 & 0.98322 & 0.000341 \\
 10 & 1.23951 & 1.23949 & 0.000022 \\
 15 & 1.41265 & 1.41264 & 0.000001\\
 20 & 1.54177 & 1.54177 & \approx 0 \\
 25 & 1.64453 & 1.64453 & \approx 0 \\
 30 & 1.72983 & 1.72983 & \approx 0
\end{array}
\right)$$
A: $$\lim_{n \to \infty}\sum_{k=n}^{n^2}{\frac{1}{2^k}}=0$$
It works with every converging sum
