Alternating series Guide, Meaning , Facts, Information and Description
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 the
natural logarithm of 2.
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 the natural logarithm of 2:
One possible reordering for this series is as follows (the only purpose of
the brackets in the first line is to help clarity):
-
-
-
A proof of this assertion runs along the lines: the
greedy algorithm for σ is
correct.
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