Home > Calculus > Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series

How do you use the Alternating Series Test?

Answer 1

Alternating Series Test states that an alternating series of the form #sum_{n=1}^infty (-1)^nb_n#, where #b_n ge0#, converges if the following two conditions are satisfied:

Let us apply the test to the alternating series below.

#sum_{n=1}^infty(-1)^{n-1}1/sqrt{n}#

In this series, #b_n=1/sqrt{n}#.

Let us check the two conditions.

Hence, we conclude that the alternating series converges.

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Answer 2

Cameron Alexander

To use the Alternating Series Test:

Determine if the series is alternating, meaning the terms alternate in sign (positive, negative, positive, etc.).
Check if the absolute value of the terms decreases as n increases.
Verify that the limit of the absolute value of the terms as n approaches infinity is zero.
If these conditions are met, then the series converges according to the Alternating Series Test.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.