# Thread: [SOLVED] prove lim (1-1/n)^n = 1/e

1. ## [SOLVED] prove lim (1-1/n)^n = 1/e

Prove $\lim_{n\to\infty}\left(1-\frac{1}{n}\right)^n=\frac{1}{e}$.

Solution:

We begin with the identity

$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n=e$.

From this, we may see that

$\lim_{n\to\infty}\left(\frac{n+1}{n}\right)^n=e$.

With this in mind, let's go back to the original limit in question:

$\lim_{n\to\infty}\left(1-\frac{1}{n}\right)^n=\lim_{n\to\infty}\left(\frac{ n-1}{n}\right)^n$

$=\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n\lef t(\frac{(n-1)(n+1)}{n^2}\right)^n$

$=\lim_{n\to\infty}\frac{\left(\frac{(n-1)(n+1)}{n^2}\right)^n}{\left(\frac{n+1}{n}\right) ^n}$

It seems like I'm getting close, here, but I don't know enough about nth roots to compute this limit.

Any ideas?

2. Oh, geez. It took me a week, but I finally see it now, and I wonder how I missed something so obvious.

Thanks for the help!

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# proof for 1/e

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