Multiple Partial Fractions

Algebra Level 4

Given that \(k\in\mathbb{Z}^+\) is a constant and the expression of \[\frac{x^3+x^2+x+k}{(x+k)^2 (x^2+k^2)}\]

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

where \(C_{k,1}, C_{k,2}, C_{k,3}, C_{k,4}\) are constants, determine the value of \[\sum_{k=2}^{50} {C_{k,2}}\]

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