TIFR 2013 Problem 40 Solution -Convergence of alternating series

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.
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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