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Question Number 103410 by Ar Brandon last updated on 14/Jul/20
Σ_(k=0) ^n (((−1)^k )/(k!))=?
$$\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} }{\mathrm{k}!}=? \\ $$
Answered by mathmax by abdo last updated on 15/Jul/20
we have e^(−1) =Σ_(k=0) ^∞ (((−1)^k )/(k!)) =Σ_(k=0) ^n (((−1)^k )/(k!)) +Σ_(k=n+1) ^∞ (((−1)^k )/(k!)) =Σ_(k=0) ^n (((−1)^k )/(k!)) +R_n ( rest at the serie of taylor) ⇒ Σ_(k=0) ^n (((−1)^k )/(k!)) =(1/e)−R_n with lim_(n→+∞) R_n =0
$$\mathrm{we}\:\mathrm{have}\:\mathrm{e}^{−\mathrm{1}} \:=\sum_{\mathrm{k}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} }{\mathrm{k}!}\:=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} }{\mathrm{k}!}\:+\sum_{\mathrm{k}=\mathrm{n}+\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} }{\mathrm{k}!} \\ $$$$=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} }{\mathrm{k}!}\:+\mathrm{R}_{\mathrm{n}} \:\:\:\left(\:\mathrm{rest}\:\mathrm{at}\:\mathrm{the}\:\mathrm{serie}\:\mathrm{of}\:\mathrm{taylor}\right)\:\Rightarrow \\ $$$$\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} }{\mathrm{k}!}\:=\frac{\mathrm{1}}{\mathrm{e}}−\mathrm{R}_{\mathrm{n}} \:\:\:\mathrm{with}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\mathrm{R}_{\mathrm{n}} =\mathrm{0} \\ $$

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