<s>[INST] How to evaluate the sum : $\sum_{k=1}^{n} \frac{k}{k^4+1/4}$ I have been trying to figure out how to evaluate the following sum:
$$S_n=\sum_{k=1}^{n} \frac{k}{k^4+1/4}$$
In the problem, the value of $S_{10}$ was given as $\frac{220}{221}$. 
I have tried  partial decomposition, no where I go. Series only seems like it telescopes, otherwise there isn't another way.
Any ideas are appreciated! 
 [/INST] Try to break the denominator into product of two factors:
$$\begin{align}
4k^4 + 1 &= (2k^2)^2 + 1 + 2 (2k^2) - 2 (2k^2) \\
&= (2k^2 +1)^2 - (2k)^2 \\
&= (2k^2 +2k +1)(2k^2 -2k+1)
\end{align}$$
Using this we see the general term as:
$$T_k =  \dfrac{1}{2k^2-2k+1} - \dfrac{1}{2k^2+2k+1} \\
T_{k+1} =  \dfrac{1}{2k^2+2k+1} - \dfrac{1}{2(k+1)^2+2(k+1)+1} $$
Alternate terms cancel and the sum telescopes to:
$$1-\frac{1}{2n^2+2n+1}$$
</s>