TIFR 2013 Problem 32 Solution - Limit of a Sequence

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.

Also Visit: College Mathematics Program of Cheenta

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