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Math Help - help proof limit n to infinity (n+1)^1/3 - n (1/3)

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    help proof limit n to infinity (n+1)^1/3 - n ^(1/3)

    Please help to shed some light on how to proof this. This is a home work problem from my real analysis class.

    thanks
    Last edited by kanli; October 11th 2012 at 10:01 PM.
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    Re: help proof limit n to infinity (n+1)^1/3 - n (1/3)

    Last edited by MaxJasper; October 12th 2012 at 09:32 AM.
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    Re: help proof limit n to infinity (n+1)^1/3 - n (1/3)

    Quote Originally Posted by MaxJasper View Post
    Try (n+1)/n = 1+1/n
    How do I do that? I can't find a common factor that will turn it into the form (n+1). I have tried to multiply with [(n+1)^1/3 + n^(1/3)] / [(n+1)^1/3 + n^(1/3)] . But can't get rib of the power.
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    Lightbulb Re: help proof limit n to infinity (n+1)^1/3 - n (1/3)

    Last edited by MaxJasper; October 12th 2012 at 09:31 AM.
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    Re: help proof limit n to infinity (n+1)^1/3 - n ^(1/3)

    Quote Originally Posted by kanli View Post
    Please help to shed some light on how to proof this. This is a home work problem from my real analysis class.

    thanks
    Is this supposed to be \displaystyle \begin{align*} (n + 1)^{\frac{1}{3}} - n^{\frac{1}{3}} \end{align*}?
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    Re: help proof limit n to infinity (n+1)^1/3 - n ^(1/3)

    Quote Originally Posted by kanli View Post
    Please help to shed some light on how to proof this. This is a home work problem from my real analysis class.

    thanks
    Hint: Use the difference of cubes formula

    (a-b)(a^2+ab+b^2)=a^3-b^3

    This gives the limit

    \lim_{x \to \infty}\frac{(n+1)-n}{(n+1)^{\frac{2}{3}}+n^{\frac{1}{3}} (n+1)^{\frac{2}{3}}+n^{\frac{2}{3}} }
    Thanks from kanli
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    Re: help proof limit n to infinity (n+1)^1/3 - n ^(1/3)

    thank you very much. I solved the problem.
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