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

• Oct 11th 2012, 08:52 PM
kanli
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
• Oct 11th 2012, 08:57 PM
MaxJasper
Re: help proof limit n to infinity (n+1)^1/3 - n (1/3)
(Giggle)
• Oct 11th 2012, 09:05 PM
kanli
Re: help proof limit n to infinity (n+1)^1/3 - n (1/3)
Quote:

Originally Posted by MaxJasper
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.
• Oct 11th 2012, 09:15 PM
MaxJasper
Re: help proof limit n to infinity (n+1)^1/3 - n (1/3)
(Itwasntme)
• Oct 12th 2012, 03:22 AM
Prove It
Re: help proof limit n to infinity (n+1)^1/3 - n ^(1/3)
Quote:

Originally Posted by kanli
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*}?
• Oct 12th 2012, 04:31 AM
TheEmptySet
Re: help proof limit n to infinity (n+1)^1/3 - n ^(1/3)
Quote:

Originally Posted by kanli
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

$\lim_{x \to \infty}\frac{(n+1)-n}{(n+1)^{\frac{2}{3}}+n^{\frac{1}{3}} (n+1)^{\frac{2}{3}}+n^{\frac{2}{3}} }$
• Oct 13th 2012, 05:02 PM
kanli
Re: help proof limit n to infinity (n+1)^1/3 - n ^(1/3)
thank you very much. I solved the problem.