The Alternating series reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Alternating series

In mathematics, an alternating series is an infinite series of the form

with a_n ≥ 0. A sufficient condition for the series to converge is that it converges absolutely. But this is often too strong a condition to ask: it is not necessary. For example, the harmonic series

diverges, while the alternating version

converges to .

A broader test for convergence of an alternating series is the Cauchy criterion: if the sequence is monotone decreasing and tends to zero, then the series

converges.

A conditionally convergent series is an infinite series that converges, but does not converge absolutely. The following non-intuitive result is true: if the real series

converges conditionally, then for every real number there is a reordering of the series such that

As an example of this, consider the series above for :

One possible reordering for this series is as follows (the only purpose of the brackets in the first line is to help clarity):

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