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Find a real set on which this series is convergent

Source: 2009 József Wildt International Math Competition

April 17, 2020
limitsseries

Problem Statement

Let the series s(n,x)=∑k=0n(1−x)(1−2x)(1−3x)⋯(1−nx)n!s(n,x)=\sum \limits_{k= 0}^n \frac{(1-x)(1-2x)(1-3x)\cdots(1-nx)}{n!} Find a real set on which this series is convergent, and then compute its sum. Find also lim⁡(n,x)→(∞,0)s(n,x)\lim \limits_{(n,x)\to (\infty ,0)} s(n,x)
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