<s>[INST] Find expression for : $ S_n =\sum_{i=1}^{n} \frac{i}{i^4+i^2+1} $ I want to find a formula for the sum of this series using its general term.
How to do it?
Series
$$
S_n = \underbrace{1/3 + 2/21 + 3/91 + 4/273 + \cdots}_{n \text{ terms}}
$$
General Term
$$
S_n = \sum_{i=1}^{n} \frac{i}{i^4+i^2+1}
$$
 [/INST] Set $$\color{blue}{a_k=  \frac{1}{k^2 -k +1} \implies a_{k+1}=  \frac{1}{(k+1)^2 -k } =\frac{1}{k^2+ k +1}} $$
But since $ (k^2 -k +1)(k^2 +k +1) =k^4 +k^2 +1$ we have,
$$ a_{k+1} -a_k = \frac{1}{k^2 +k +1}-  \frac{1}{k^2 -k +1} = \frac{2k}{(k^2 -k +1)(k^2 +k +1)} =\frac{2k}{k^4 +k^2 +1}$$
Then 
$$a_n-a_0 = \sum_{k=0}^{n-1}a_{k+1} -a_k=2\sum_{k=0}^{n-1}\frac{k}{k^4 +k^2 +1} $$
</s>