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Thread: Limits and sequences

  1. #1
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    Limits and sequences

    Is this a standard result?

    $\displaystyle \displaystyle\lim_{n\to\infty}a_n^{n}=\left( \lim_{n\to\infty}a_n\right)^n$

    Because, if I'm given this question:

    Find

    $\displaystyle \displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n}\ right)^{4n}$

    It would equal to

    $\displaystyle \displaystyle\lim_{n\to\infty}\left[\left(1+\frac{1}{n}\right)^n\left(1+\frac{1}{n}\ri ght)^n\left(1+\frac{1}{n}\right)^n\left(1+\frac{1} {n}\right)^n\right]$

    $\displaystyle \displaystyle\left[\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n\righ t]^4$
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  2. #2
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    I believe it is correct, because the limit of a product is the same as the product of the limits, and exponentiation is "repeated multiplication".
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  3. #3
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    But the problem is NOT $\displaystyle \lim_{n\to\infty}\left(a_n\right)^n$, it is $\displaystyle \lim_{n\to\infty}\left(a_n\right)^4$ and that definitely is $\displaystyle \left(\lim_{n\to\infty} a_n\right)^4$.
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