Alternating series test

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The alternating series test is a method used to prove that infinite series of terms converge. It was discovered by Gottfried Leibniz and is sometimes known as Leibniz's test or the Leibniz criterion.

A series of the form

where all the a_n are non-negative, is called an alternating series. If the sequence a_n approaches 0 as n approaches infinity, and the sequence a_n is monotone decreasing (i.e. each a_n is smaller than a_n−1), then the series converges. If L is the sum of the series,

then the partial sum

approximates L with error

It is perfectly possible for a series to have its partial sums S_k fulfill this last condition without the series being alternating. For a straightforward example, consider:

Proof

^[1]

We are given a series of the form . The limit of the sequence equals 0 as approaches infinity, and each is smaller than (i.e. the sequence is monotone decreasing).

Proof of convergence

The (2n+1)-th partial sum of the given series is . As every sum in brackets is non-positive, and as , then the (2n+1)-th partial sum is not greater than .

That very (2n+1)-th partial sum can be written as . Every sum in brackets is non-negative. Therefore, the series is monotonically increasing: for any the following holds: .

From the two paragraphs it follows by the monotone convergence theorem that there exists such a number s that .

As and as , then . The sum of the given series is , where is a finite number. Thus, convergence is proved.

Another way to prove this is showing that the sequence of partial sums are a cauchy sequence.

Proof of partial sum error

In the proof of convergence we saw that is monotonically increasing. Since , and every term in brackets is non-positive, we see that is monotonically decreasing. By the previous paragraph, , hence . Similarly, since is monotonically increasing and converging to , we have . Hence we have for all n.

Therefore if k is odd we have , and if k is even we have .

See also

• Alternating series
• Dirichlet's test

Literature

• Knopp, Konrad, "Infinite Sequences and Series", Dover publications, Inc., New York, 1956. (§ 3.4) ISBN 0-486-60153-6

• Whittaker, E. T., and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1963. (§ 2.3) ISBN 0-521-58807-3

• Last, Philip, "Sequences and Series", New Science, Dublin, 1979. (§ 3.4) ISBN 0-286-53154-3

References

1. ^ Beklemishev, Dmitry V. (2005). Analytic geometry and linear algebra course (10 ed.). FIZMATLIT. 

External links

• Weisstein, Eric W., "Leibniz Criterion" from MathWorld.