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Math Help - Limit proof

  1. #1
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    Limit proof

    Prove that  \lim _{n \rightarrow \infty } n^ { \frac {1}{n} } = 1

    Proof so far.

    Define the sequence  \{ a_n \} = n^ { \frac {1}{n} } -1

    I want to show that  \{ a_n \} \rightarrow 0
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by tttcomrader View Post
    Prove that  \lim _{n \rightarrow \infty } n^ { \frac {1}{n} } = 1

    Proof so far.

    Define the sequence  \{ a_n \} = n^ { \frac {1}{n} } -1

    I want to show that  \{ a_n \} \rightarrow 0

    Take logs and show that

     \lim _{n \to \infty } \frac {\ln(n)}{n} = 0

    You can do this by switching to a continuous variable and then using L'Hopital's rule

    RonL
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  3. #3
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    Quote Originally Posted by tttcomrader View Post
    Prove that  \lim _{n \rightarrow \infty } n^ { \frac {1}{n} } = 1

    Proof so far.

    Define the sequence  \{ a_n \} = n^ { \frac {1}{n} } -1

    I want to show that  \{ a_n \} \rightarrow 0
    First a_n \geq 0. Second (a_n + 1)^n = n.
    Therefore, n = (a_n+1)^n \geq  1 + na_n + \tfrac{1}{2}n(n-1)a_n^2 \geq \tfrac{1}{2}n(n-1)a_n^2

    Thus, 0 \leq a_n \leq \sqrt{\frac{2}{n-1}}

    Now use squeeze theorem to show \lim ~ a_n = 0.
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