<s>[INST] Series convergence without sigma notation Consider the following series:
$$\frac{1}{1} + \frac{10}{2} + \frac{100}{3} - \frac{37}{4} - \frac{37}{5} - \frac{37}{6} + \frac{1}{7} + \frac{10}{8} + \frac{100}{9} - \frac{37}{10} - \frac{37}{11} - \frac{37}{12} + \dots$$
This series seems to converge but I am not able to prove it. It seems impossible to write thing whole thing into summation notation. I tried to group by modulo 6, but then for example sum of $\frac{1}{6n+1}$ diverges already so that doesn't seem right. Also tried to change the 37s into powers of 10 to maybe use the comparison test but also failed.
Any suggestions?
 [/INST] Added for your curiosity but too long for a comment.
Starting from Robert Israel's elegant answer, it is possible to have quite accurate approximations of the partiel sums using the fact that, for large values of $p$ we have
$$\Psi (p)=\log \left({p}\right)-\frac{1}{2 p}-\frac{1}{12 p^2}+\frac{1}{120
   p^4}+O\left(\frac{1}{p^6}\right)$$ making
$$S_m=\left(\frac{64}{3} \ln(2) - \frac{63}{4} \ln(3) +\frac{161}{36} \pi \sqrt{3}\right)-\frac{13}{2
   m}+\frac{23}{3 m^2}-\frac{311}{36 m^3}+\frac{499}{54 m^4}-\frac{36377}{3888
   m^5}+O\left(\frac{1}{m^6}\right)$$ Below are listed some values
$$\left(
\begin{array}{ccc}
 m & \text{approximation} & \text{exact} \\
 2 & 19.691108 & 19.787015 \\
 3 & 20.259944 & 20.269370 \\
 4 & 20.565282 & 20.567065 \\
 5 & 20.768485 & 20.768971 \\
 6 & 20.914700 & 20.914867 \\
 7 & 21.025135 & 21.025202 \\
 8 & 21.111527 & 21.111558 \\
 9 & 21.180965 & 21.180981
\end{array}
\right)$$
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