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Problem:True/False


\( \lim_{n\to \infty } (n+1)^{1/3} -n^{1/3} = \infty \)


Hint:

Simplify the given expression.


Discussion:

We feel that \( (n+1)^{1/3} \) goes to infinity at the same speed as \( n^{1/3} \). So in fact, the above limit should be zero.

We make this little bit more rigorous.

\(   (n+1)^{1/3} -n^{1/3} = \frac{n+1-n}{(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3} } \)

\(  =\frac{1}{(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3} } \to 0\) as \(n\to \infty \).


Helpdesk

  • What is this topic: Real Analysis
  • What are some of the associated concept: Limit of a Sequence
  • Book Suggestions: Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert