April's Lagrange Interpolation

Algebra Level 5

Let \(P(x)\) be a monic polynomial of degree \(2015\), satisfying, for \(x\in \mathbb{Z}^+, x\le 2015\), \[P(x)=\sum_{k=1}^x \dfrac{Q(k)^3+1}{k^3+1}\] where \[Q(x)=\left\{ \begin{array}{l}-\sum\limits_{i=1}^{\infty}\dfrac{1}{i^{|x|}}\qquad \text{if the sum converges}\\ -\dfrac{1}{|x|}\qquad \text{if the sum diverges}\end{array}\right.\] Now, let \(a_n\) be the coefficient of the \(x^n\) term in \(P(x)\). Finally, let \[S=\sum_{i=1}^{2015}a_{i-1}\] Find the positive remainder when \(\lfloor S\rfloor^{2015-\lceil S\rceil}\) is divided by \(1000\).

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