300 Followers Problem - Polynomial Differential Reciprocal Summations!

Algebra Level 5

Let P(x)=z=02015azxz\large{P(x) = \displaystyle \sum_{z=0}^{2015} a_z x^z} be a polynomial with real coefficients aza_z defined by az=ln(e+z)a_z = \ln(e+z) having simple roots r1,r2,,r2015r_1, r_2, \ldots, r_{2015} . Evaluate the value of the following upto three correct places of decimals:

k=02014(i=12015rik1P(ri))= ?\large{\sum_{k=0}^{2014} \left(\sum_{i=1}^{2015} \dfrac{r_i^{k-1}}{P'(r_i)} \right) = \ ?}

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