Alternating Series Test

If , then an alternating series

converges if both of the following conditions are satisfied:

1)

2) is a decreasing sequence; that is, for all n.

Note: This does not say that if , the series diverges by the Alternating Series

Test. The Alternating Series Test can only be used to prove convergence.

If , then the series diverges by the n th Term Test for Divergence,

not by the Alternating Series Test.

Definitions:

is absolutely convergent if converges.

is conditionally convergent if converges but diverges.

Suppose s_n has a mix of positive and negative signed terms.

Ex: s_n = + …

If all of the signs were positive, the sum would be higher because there would not be as much cancellation. If all of the signs were positive, then the series would converge by the p-series test. So it is true that this series also converges because it has the “advantage” of cancellation. This means that the given series converges absolutely (because it still would converge without the negative signs).

If the signs of a series (a) are strictly alternating (not just some random term shave negative signs), (b) decrease in absolute value, and (c) go toward zero, then the series converges (and the error when you stop the series is contained in the first unused term). This is called Leibniz’ Theorem.

Ex: s_n = + … converges (even though the harmonic series diverges!)

Ex: s_n = 12 – 6 + 3 - converges

(Note: s_n = 12 – 6 + 3 - converges because it is a geometric series with | r | < 1, so s_n = )

If s_n converges with its positive and negative signs but it would diverge if all of the signs were the same (such as all positive), then the series convergesconditionally (and not absolutely).

Ex: s₁ = converges by Leibniz’ Theorem but

s₂ = diverges by the p-series

So s₁ converges conditionally.

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