Alternating series
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In mathematics, an alternating series is an infinite series of the form
- <math>\sum_{n=0}^\infty (-1)^n\,a_n,<math>
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
- <math>\sum_{n=0}^\infty \frac{1}{n+1},<math>
diverges, while the alternating version
- <math>\sum_{n=0}^\infty \frac{(-1)^n}{n+1}<math>
converges to the natural logarithm of 2.
A broader test for convergence of an alternating series is the Cauchy criterion: if the sequence <math>a_n<math> is monotone decreasing and tends to zero, then the series
- <math>\sum_{n=0}^\infty (-1)^n\,a_n<math>
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
- <math>\sum_{n=0}^\infty (-1)^n\,a_n<math>
converges conditionally, then for every real number <math>\beta<math> there is a reordering <math>\sigma<math> of the series such that
- <math>\sum_{n=0}^\infty (-1)^{\sigma(n)}\,a_{\sigma(n)}=\beta.<math>
As an example of this, consider the series above for the natural logarithm of 2:
- <math>\ln 2=\sum_{n=0}^\infty \frac{(-1)^n}{n+1}=1-\frac12+\frac13-\frac14+\frac15-\cdots.
<math>
One possible reordering for this series is as follows (the only purpose of the brackets in the first line is to help clarity):
- <math>1-\frac12-\frac14+\left(\frac13-\frac16\right)-\frac18+\left(\frac15-\frac1{10}\right)-\frac1{12}
+\left(\frac17-\frac1{14}\right)-\frac1{16}+\cdots<math>
- <math>=\frac12-\frac14+\frac16-\frac18+\frac1{10}-\cdots<math>
- <math>=\frac12\left(1-\frac12+\frac13-\frac14+\frac15-\cdots\right)<math>
- <math>=\frac12\,\ln2.<math>
A proof of this assertion runs along the lines: the greedy algorithm for σ is correct.