Alternating series test

The alternating series test is a method used to test infinite series of terms for convergence. It was developed by Gottfried Leibniz and is sometimes known as Leibniz's test.

A series of the form

<math>\sum_{n=1}^\infty a_n(-1)^n<math>

with a_n ≥ 0 is called an alternating series. The series will converge if the sequence a_n is monotone decreasing and converges to 0. The converse is in general not true.

A simple example is the series

<math>\sum_{n=1}^\infty \frac{(-1)^n}{n} = -1+\frac{1}{2}-\frac{1}{3}\ldots<math>

which converges to the value ln(1/2) while

<math>\sum_{n=1}^\infty \frac{1}{n} = 1+\frac{1}{2}+\frac{1}{3}\ldots<math>

is divergent.

References

• Knopp, Konrad, "Infinite Sequences and Series", Dover publications, Inc., New York, 1956. (§ 3.4) ISBN 0486601536

• Whittaker, E. T., and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1963. (§ 2.3) ISBN 0521588073