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:\sum_{n=0}^\infty (-1)^n\,a_n,

with ''an'' ≥ 0. A finite sum of this kind is an alternating sum.

A ''sufficient'' condition for the

:\sum_{n=0}^\infty rac{1}{n+1},

diverges, while the alternating version

:\sum_{n=0}^\infty rac{(-1)^n}{n+1}

converges to the Natural Logarithm of 2.

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

:\sum_{n=0}^\infty (-1)^n\,a_n

converges.

The Partial Sum

:s_n = \sum_{k=0}^n (-1)^k a_k

can be used to approximate the sum of a convergent alternating series.
If a_n is monotone decreasing and tends to zero, then the error
in this approximation is less than a_{n+1}.

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

:\sum_{n=0}^\infty (-1)^n\,a_n

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

:\sum_{n=0}^\infty (-1)^{\sigma(n)}\,a_{\sigma(n)}=\beta.

As an example of this, consider the series above for the natural logarithm of 2:

:\ln 2=\sum_{n=0}^\infty rac{(-1)^n}{n+1}=1- rac12+ rac13- rac14+ rac15-\cdots.


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

:1- rac12- rac14+\left( rac13- rac16 ight)- rac18+\left( rac15- rac1{10} ight)- rac1{12}
+\left( rac17- rac1{14} ight)- rac1{16}+\cdots
:= rac12- rac14+ rac16- rac18+ rac1{10}-\cdots
:= rac12\left(1- rac12+ rac13- rac14+ rac15-\cdots ight)
:= rac12\,\ln2.

A proof of this assertion runs along the lines: the Greedy Algorithm for σ is Correct .


SEE ALSO