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Math Help - easy convergence question

  1. #1
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    easy convergence question

    This question seems like it should be so easy, but I just can't quite get there:

    Use the fact that (1 + \frac{1}{n})^n \rightarrow e to deduce that (1 - \frac{1}{n})^n \rightarrow \frac{1}{e}.

    So far this is what I have but I have a feeling i'm going about it the wrong way:
    (\frac{n+1}{n})^n \rightarrow e
    By algebra of limits: (\frac{n}{n+1})^n \rightarrow \frac{1}{e}
    So (1- \frac{1}{n+1})^n \rightarrow \frac{1}{e}.
    But from here I can't see how I can get the answer required.
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  2. #2
    Senior Member
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    Substitute n = -m and simplify (if you know the limit holds for positive and negative n).

    For your solution
    (1- \frac{1}{n+1})^n = (1- \frac{1}{n+1})^{n+1}(1 - \frac{1}{n+1})^{-1} = (1- \frac{1}{n+1})^{n+1}(\frac{n+1}{n})
    Now use the substitution n = m - 1
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  3. #3
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    Cheers
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