Sum of an infinite series of fractions involving multiple terms in the denominator This is the series in question:
$$S = \frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4} \ldots$$
The general term seems to be:
$$T_n=\frac{n}{1+n^2+n^4}$$
In the original question, which is from Resonance DLPD Algebra, the value of $14S$ is asked.
I haven't been able to make any progress on this question even after trying for a long time. I don't know how to handle series where the denominator contains sum of numbers in general. Can you please provide a hint?

We are looking for: $S_n=\sum\limits_{n=1}^\infty\frac{n}{1+n^2+n^4}$
Let's realize that $\frac{n}{1+n^2+n^4}=\frac{1}{2}\big(\frac{1}{1-n+n^2}-\frac{1}{1+n+n^2}\Big)=\frac{1}{2}\big(\frac{1}{(n-\frac{1}{2})^2+\frac{3}{4}}-\frac{1}{(n+\frac{1}{2})^2+\frac{3}{4}}\Big)$
Which provide telescopic sum so $S=\frac{1}{2}$
