Question Number 205574 by Red1ight last updated on 24/Mar/24
$${S}=\frac{\mathrm{1}}{\mathrm{2}!}−\frac{\mathrm{1}}{\mathrm{3}!}+\frac{\mathrm{1}}{\mathrm{4}!}−\frac{\mathrm{1}}{\mathrm{5}!}... \\ $$$${S}=? \\ $$
Commented by Frix last updated on 25/Mar/24
$${S}=\frac{\mathrm{1}}{\mathrm{e}} \\ $$
Answered by mathzup last updated on 25/Mar/24
$${S}=\sum_{{n}=\mathrm{2}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}=\sum_{{n}=\mathrm{0}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}−\mathrm{1}−\left(−\mathrm{1}\right) \\ $$$$=\frac{\mathrm{1}}{{e}}\:\:\:\:\:\:\:\:\:\left(\sum_{{n}=\mathrm{0}} ^{\infty} \frac{{x}^{{n}} }{{n}!}={e}^{{x}} \:\:\right) \\ $$$$ \\ $$