Multiple Partial Fractions

Algebra Level 4

\[\large \frac{x^3+x^2+x+k}{(x+k)^2 (x^2+k^2)}\]

Given that \(k\in\mathbb{Z}^+\) is a constant and the expression above decomposes into partial fractions of

\[\frac{C_{k,1}}{x+k}+\frac{C_{k,2}}{(x+k)^2}+\frac{C_{k,3}x+C_{k,4}}{x^2+k^2}\]

where \(C_{k,1}\), \(C_{k,2}\), \(C_{k,3}\), and \(C_{k,4}\) are constants. Determine the value of \(\displaystyle \sum_{k=2}^{50} {C_{k,2}}-25\).

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