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Convergence of alternating series (TIFR 2013 problem 40)

Question:

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.

September 24, 2017

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