Alternating series: Difference between revisions
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:<math>\sum_{n=0}^\infty (-1)^n\,a_n,</math> |
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with ''a<sub>n</sub>'' ge; 0. A ''sufficient'' condition for the [[series (mathematics)|series]] to converge is that it [[absolute convergence|converges absolutely]]. But this is often too strong a condition to ask:it is not ''necessary''. For example, the [[harmonic series (mathematics)|harmonic series]] |
with ''a<sub>n</sub>'' ≥ 0. A ''sufficient'' condition for the [[series (mathematics)|series]] to converge is that it [[absolute convergence|converges absolutely]]. But this is often too strong a condition to ask:it is not ''necessary''. For example, the [[harmonic series (mathematics)|harmonic series]] |
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:<math>\sum_{n=0}^\infty \frac1n,</math> |
:<math>\sum_{n=0}^\infty \frac1n,</math> |
Revision as of 10:30, 29 February 2004
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