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Math Help - Limit Of Sequence

  1. #1
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    Limit Of Sequence

    Does someone mind showing me how to compute this limit algebraically? I'm pretty sure it goes to e, but I can't figure how to show it. Thanks for your help.

    lim (n+1)^(1/ln(n+1))
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  2. #2
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    Quote Originally Posted by jamesHADDY View Post
    Does someone mind showing me how to compute this limit algebraically? I'm pretty sure it goes to e, but I can't figure how to show it. Thanks for your help.

    lim (n+1)^(1/ln(n+1))
    \mathop {\lim }\limits_{n \to \infty } (n + 1)^{1/\ln (n + 1)}.

    Substitute u = \frac{1}<br />
{{\ln (n + 1)}}, the limit becomes

    \mathop {\lim }\limits_{u \to 0} \left( {e^{1/u} } \right)^u  = \mathop {\lim }\limits_{u \to 0} e.

    The rest follows.
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  3. #3
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    Do you mind elaborating? I don't see how this substitution equals the original and how you get the limit as u goes to infinity?
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    Quote Originally Posted by jamesHADDY View Post
    Do you mind elaborating? I don't see how this substitution equals the original and how you get the limit as u goes to infinity?
    clearly \left( e^{1/u} \right)^u = e, so the result follows if we take the limit as u goes to zero. i suppose you are wondering where e^{1/u} came from?

    note that e^{\ln x} = x

    thus: n + 1 = e^{\ln (n + 1)}, but since u = \frac 1{\ln (n + 1)}, we have that \ln (n + 1) = \frac 1u, thus it follows that e^{\ln (n + 1)} = e^{1/u}
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