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
$$
S_n = \sum_{i=1}^{n} \frac{i}{i^4+i^2+1}
$$

Write it out:
$$
\begin{eqnarray}
\sum_{i=1}^\infty \frac{i}{i^4 + i^2 + 1}
&=& \sum_{i=1}^\infty \left( \frac{2}{4i^2 - 4i + 4}
  - \frac{2}{4i^2 + 4i + 4}\right)\\
&=&  \sum_{i=1}^\infty \left( \frac{2}{\Big(2i-1\Big)^2 + 3}
  - \frac{2}{\Big( 2i + 1\Big)^2 + 3}\right)\\
&=&  \frac{2}{\Big( 2 - 1\Big)^2 + 3} + \sum_{i=1}^\infty \left( \frac{2}{\Big(2i+1\Big)^2 + 3}
  - \frac{2}{\Big( 2i + 1\Big)^2 + 3}\right)\\
&=& \frac{1}{2}.
\end{eqnarray}
$$
