In a German science journal I read that PH Schoute asked this in the Educational Times around 1900:


What is the sum of


$$S = \frac{1}{1}+\frac{1}{2+3}+\frac{1}{4+5+6}+\frac{1}{7+8+9+10} +\cdots\text{?}$$


I don't know if an answer has been found. Anyone?



What's this infinite sum?
You can have a very good approximation of the partial sums writing first
$$\frac2{n(n^2+1)}=\frac 2{n(n+i)(n-i)}=\frac{2}{n}-\frac{1}{n-i}-\frac{1}{n+i}.$$
Using generalized harmonic numbers
$$S\_p=\sum\_{n=1}^p\frac2{n(n^2+1)}=2 H\_p-H\_{p-i}-H\_{p+i}+H\_i+H\_{-i}.$$
Using asymptotics
$$S\_p=\left(H\_{-i}+H\_i\right)-\frac{1}{p^2}+\frac{1}
 {p^3}-\frac{1}{p^5}+\frac{2}{3
 p^6}+\frac{1}{p^7}+O\left(\frac{1}{p^8}\right).$$ As @Robert Israel wrote in [comments](https://mathoverflow.net/questions/443247/whats-this-infinite-sum#comment1144169_443247)
$$\left(H\_{-i}+H\_i\right)=2 \gamma + 2 \text{Re}(\Psi(i)).$$


