<s>[INST] Sum of an infinite series $\sum _{k=3}^{\infty }\:\frac{1}{k\left(k^4-5k^2+4\right)^2}$ I want to compute this series, $$\sum _{k=3}^{\infty }\:\frac{1}{k\left(k^4-5k^2+4\right)^2}.$$ I don't know how to contiune after factoring, which yields $\frac{1}{k(k-1)^2(k+1)^2(k-2)^2(k+2)^2}.$
I know that $\frac{1}{k(k-1)}$ telescopes. However, I can't apply this here, as the product of the sum is not equal to the sum of the product. I only want hints for now.
 [/INST] Perform a partial fraction decomposition; specifically, find constants $A_1, A_2, \ldots, A_9$ such that $$\begin{align}\frac{1}{k(k^4-5k^2+4)^2} &= \frac{A_1}{k} + \frac{A_2}{k-1} + \frac{A_3}{k+1} + \frac{A_4}{k-2} + \frac{A_5}{k+2} \\&\quad+ \frac{A_6}{(k-1)^2} + \frac{A_7}{(k+1)^2} + \frac{A_8}{(k-2)^2} + \frac{A_9}{(k+2)^2}. \end{align}$$
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