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Question-193268




Question Number 193268 by 073 last updated on 09/Jun/23
Answered by qaz last updated on 09/Jun/23
(Σ_(n=0) ^∞ (((−x)^n )/(n!)))(Σ_(n=0) ^∞ (((−x)^n x)/(n!(n+1))))=xe^(−x) Σ_(n=0) ^∞ (((−x)^n )/(n!))∫_0 ^1 y^n dy  =xe^(−x) ∫_0 ^1 e^(−xy) dy=e^(−x) (1−e^(−x) )
$$\left(\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−{x}\right)^{{n}} }{{n}!}\right)\left(\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−{x}\right)^{{n}} {x}}{{n}!\left({n}+\mathrm{1}\right)}\right)={xe}^{−{x}} \underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−{x}\right)^{{n}} }{{n}!}\int_{\mathrm{0}} ^{\mathrm{1}} {y}^{{n}} {dy} \\ $$$$={xe}^{−{x}} \int_{\mathrm{0}} ^{\mathrm{1}} {e}^{−{xy}} {dy}={e}^{−{x}} \left(\mathrm{1}−{e}^{−{x}} \right) \\ $$

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