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Math Help - [SOLVED] prove lim (1-1/n)^n = 1/e

  1. #1
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    [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?
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  2. #2
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  3. #3
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    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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