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Question Number 193268 by 073 last updated on 09/Jun/23

Answered by qaz last updated on 09/Jun/23

$$\left(\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{{(-x)}^n}{n!}\right)\left(\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{{(-x)}^{n}x}{n!(n+1)}\right)={xe}^{-x}\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{{(-x)}^n}{n!}{\int }_0^1{y}^{n}dy$$$$={xe}^{-x}{\int }_0^1{e}^{-xy}dy=e^{-x}(1-e^{-x})$$

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