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Math Help - interesting limit

  1. #1
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    interesting limit

    For all of the following prove without using l'Hopital's rule.

    i) Does \displaystyle\lim_{n\to\infty}\left(1+\frac{1}{\sq  rt{n}}\right)^n converge? If so what is its limit? If not prove that it diverges.

    ii) For what values p\in\mathbb{R} does \displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n^p  }\right)^n converge? If possible give a formula for the converging value.
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  2. #2
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    Quote Originally Posted by putnam120 View Post
    For all of the following prove without using l'Hopital's rule.

    i) Does \displaystyle\lim_{n\to\infty}\left(1+\frac{1}{\sq  rt{n}}\right)^n converge? If so what is its limit? If not prove that it diverges.

    ii) For what values p\in\mathbb{R} does \displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n^p  }\right)^n converge? If possible give a formula for the converging value.
    Maybe this will help.
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    thanks. I solved them by using the binomial theorem but that way works also. thanks once more.
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  4. #4
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    Quote Originally Posted by putnam120 View Post
    For all of the following prove without using l'Hopital's rule.

    i) Does \displaystyle\lim_{n\to\infty}\left(1+\frac{1}{\sq  rt{n}}\right)^n converge? If so what is its limit? If not prove that it diverges.

    ii) For what values p\in\mathbb{R} does \displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n^p  }\right)^n converge? If possible give a formula for the converging value.
    You need to know that:

    <br />
\lim_{x \to \infty}\left(1+\frac{1}{x}\right)^x = e<br />

    Then for x large enough:

    <br />
\left(1+\frac{1}{x}\right)^x=e+b_n<br />

    and |b_n|<1.

    So:

    <br />
(e-1)^{\sqrt{n}}<\left(1+\frac{1}{\sqrt{n}}\right)^n< (e+1)^{\sqrt{n}}<br />

    and the rest of part (i)should follow from the squeeze theorem.

    RonL
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