# 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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