A Family of Infinite Series (Solved)

Summary: Let $S_N := \sum_{k=0}^{\infty} (k/e)^k/(k+N)!$. Prove that $S_N = P_N(e)$, where $P_N$ is a rational polynomial in $e$ of degree $N$, and determine the coefficients of $P_N$.

Classification: Primary, classical analysis; Secondary, sequences and series

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