300 Followers Problem - Polynomial Differential Reciprocal Summations!

Algebra Level 5

Let \(\large{P(x) = \displaystyle \sum_{z=0}^{2015} a_z x^z}\) be a polynomial with real coefficients \(a_z\) defined by \(a_z = \ln(e+z) \) having simple roots \(r_1, r_2, \ldots, r_{2015} \). Evaluate the value of the following upto three correct places of decimals:

\[\large{\sum_{k=0}^{2014} \left(\sum_{i=1}^{2015} \dfrac{r_i^{k-1}}{P'(r_i)} \right) = \ ?}\]

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