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## IntroductionWith the exceptions of geometric series, where $r$ may be negative, or the rare series with telescoping partial sums, the convergence tests we have worked with so
far only work with positive-termed series. When
the terms in a series can be positive or negative, things get more
complicated; the sequence {$s_n$} of partial sums may not be
monotonic, so it can be bounded yet divergent. This module
will introduce the Alternating Series
Test, which works on series in which the terms have
alternating signs.
## Alternating Series and the Alternating Series TestAn
In other words, if the absolute values of
the terms of an alternating series are non-increasing and
converge to zero, the series converges. This is easy to test; we like alternating series. To see how easy the AST is to implement, DO:
Use the AST to see if $\displaystyle\sum_{n=1}^\infty
(-1)^{n-1}\frac{1}{n}$ converges. This series is
called the alternating harmonic series. In order to
show a series diverges, you must use another test. The best
idea is to first test an alternating series for divergence using the
Divergence Test. If the terms do not converge to zero, you are
finished. If the terms do go to zero, you are very likely to
be able to show convergence with the AST.This is a convergence-only test. Warning: The converse of
the AST is not true; we have series that are alternating and
convergent and do not satisfy the AST. For example, if we take
the terms of
$\sum\frac{1}{n^2}=1+\frac{1}{2}+\tfrac14+\tfrac19+\tfrac1{16}+\tfrac1{25}+\cdots$,
and exchange the first two terms, then the second two, etc., and
then put in alternating signs, we get
$\frac{1}{2}-1+\frac{1}{9}-\frac{1}{4}+\frac{1}{25}-\frac{1}{16}+\cdots$,
which does not satisfy the conditions of the AST since $b_{n+1}\le
b_n$ does not hold for all $n$. However, this series is
convergent (we will be able to prove its convergence later using the
ideas of Absolute Convergence).The following video will explain how the AST works, give more details on the alternating harmonic series, and look at the values of some interesting alternating series. |