TIFR 2013 Problem 32 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India’s premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.

The image is a front cover of a book named Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert. This book is very useful for the preparation of TIFR Entrance.

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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