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I want to prove that:

$\sum\limits_{i=0}^\infty \frac{(-1)^i}{i!}=\lim_{i \to \infty}\bigg(1-\frac{1}{i}\bigg)^i$

First I need to prove series is convergent. But the partial sums of the series is not monotone sequence.Can anyone tell me how to prove that it is convergent.I tried to show it is Cauchy sequence,but I couldn't. I'd appreciate any help.

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    I want to show that it is convergent without using any concepts introduced after "Convergence"topic in Analysis2012-10-26

2 Answers 2

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Use $(\rm i)$ Leibniz's Criterion, or $(\rm ii)$ a absolute convergence considerations.

$(\rm i)$ We have that $a_k=\dfrac 1 {k!}$ is monotone descreasing with $\lim\; a_k=0$, so Leibniz's criterion says $\tag{1}\sum_{k=0}^\infty (-1)^k\frac 1 {k!}$

converges.

$(\rm ii)$ $\sum_{k=0}^\infty \frac 1 {k!}=e$

so

$\tag{1}\sum_{k=0}^\infty (-1)^k\frac 1 {k!}$

converges, since it is absolutely convergent by $(1)$.

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    @Pilot I'm back. It'd be great if you added your solution.2012-10-26
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Consider series of absolute values $\sum\limits_{i=0}^n \frac{1}{i!}$ and then prove that $\sum\limits_{i=0}^{\infty} \frac{(-1)^i}{i!}$ is absolutely convergent.