TIFR 2013 Problem 40 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?

The series \(1-\frac{1}{\sqrt2}+\frac{1}{\sqrt3}-\frac{1}{\sqrt4}+…\) is divergent.

## Hint:

Recall the alternating series test (or the Leibniz test)

## Discussion:

Let \(a_n=\frac{1}{\sqrt{n}}\). The alternating series test says that if we have a series like \(a_1-a_2+a_3-a_4+…\) then a sufficient condition for the convergence of this series is: **\(a_n\) is decreasing and \(a_n\to 0 \) as \(n\to \infty \)**.

Here, \(a_n\) satisfies the above condition.

Therefore, the series converges.

## Helpdesk

**What is this topic:**Real Analysis**What are some of the associated concept:**Alternating Series, Decreasing Sequence, Convergence Criterion**Book Suggestions:**Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert