Alternating series
In mathematics, an alternating series is an infinite series of the form
with an ≥ 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