Question Number 193268 by 073 last updated on 09/Jun/23
Answered by qaz last updated on 09/Jun/23
$$\left(\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−{x}\right)^{{n}} }{{n}!}\right)\left(\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−{x}\right)^{{n}} {x}}{{n}!\left({n}+\mathrm{1}\right)}\right)={xe}^{−{x}} \underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−{x}\right)^{{n}} }{{n}!}\int_{\mathrm{0}} ^{\mathrm{1}} {y}^{{n}} {dy} \\ $$$$={xe}^{−{x}} \int_{\mathrm{0}} ^{\mathrm{1}} {e}^{−{xy}} {dy}={e}^{−{x}} \left(\mathrm{1}−{e}^{−{x}} \right) \\ $$