<s>[INST] $ \sum_{n=2}^\infty \frac{1}{n^3(n^3+1)}. $ The series is:
$$
\sum_{n=2}^\infty \frac{1}{n^3(n^3+1)}.
$$
I tried splitting the whole thing into simple fractions but I don't seem to get anywhere.
Any ideas?
 [/INST] $$\sum_{n=2}^\infty \frac{1}{n^3(n^3+1)} = \sum_{n=2}^\infty \frac{1}{n^3}+\frac{n-2}{3 (n^2-n+1)}-\frac{1}{3 (n+1)} = \\
\zeta{(3)} -1 + \sum_{n=2}^{\infty} \frac{n-2}{3 (n^2-n+1)}-\frac{1}{3 (n+1)} = \\
\zeta{(3)}-1  -\sum_{n=2}^{\infty} \frac{1}{n^3+1} \approx \\
0.015553560820970399435132949324595238582753593120862693333639...$$
</s>