Cheenta
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

TIFR 2013 Problem 11 Solution - Cauchy sequence- series


TIFR 2013 Problem 11 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 program 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


Consider the sequences \(x_n=\sum_{1}^{n} \frac{1}{j} \) and \(y_n=\sum_{1}^{n} \frac{1}{j^2} \). Then \(\left\{x_n\right\} \) is Cauchy but \(\left\{y_n \right\} \) is not.


Discussion: 


We are given sequence of partial sums of a very well known type of series.  \(\left\{x_n\right\}\) is a divergent sequence and \(\left\{y_n\right\}\) is convergent. Also, \(\mathbb{R}\) is complete. So every Cauchy sequence is convergent and any convergent sequence (as always happens in metric spaces) is Cauchy.

The true statement would be \(\left\{x_n\right\}\) is not Cauchy and \(\left\{y_n\right\}\) is Cauchy sequence.


Helpdesk

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

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com