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Math Help - Proof involving e

  1. #1
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    Proof involving e

    How do I show that for every x>0 on has
    (1+x)^(1/x)<e
    I proved that e is the limit of the equation, but I don't know how to prove it is strictly less than.

    e being defined as the limit as x tends to infinity of (1+1/x)^x.
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  2. #2
    Junior Member
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    Nov 2009
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    If f(x) = (1+x)^{1/x} is strictly decreasing on x>0 and approaches e as x \rightarrow 0, then it must be less than e for x>0. So I think showing that f'(x) < 0 for x>0 should suffice.
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