Multiple Partial Fractions

Algebra Level 4

x3+x2+x+k(x+k)2(x2+k2)\large \frac{x^3+x^2+x+k}{(x+k)^2 (x^2+k^2)}

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

Ck,1x+k+Ck,2(x+k)2+Ck,3x+Ck,4x2+k2\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 Ck,1C_{k,1}, Ck,2C_{k,2}, Ck,3C_{k,3}, and Ck,4C_{k,4} are constants. Determine the value of k=250Ck,2\displaystyle\sum_{k=2}^{50} {C_{k,2}}.

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