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S-1-2-1-3-1-4-1-5-S-




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

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