So far we have looked mainly at series consisting of positive terms,
and we have derived and used the comparison tests
and ratio test for these.
But many series have positive and negative terms, and we also need to
look at these. This page discusses a particular case of these,
*alternating series*. Some aspects of alternating series are easy
and you will see why. Other aspects are rather frightening!
To put this into context you will need to also read the pages on
absolute convergence that come later.

Definition.

For example, the series
^{
+1
}
*converges*.

Alternating Series Test.

**Proof.**

Let =_{2
}_{1}-_{2}+_{3}-_{2-1
}-_{2
}
=_{2+1
}_{1}-_{2}+_{3}-_{2-1
}-_{2
}+_{2+1
}.

**Subproof.**

By bracketing one way,

(since (_{
)
} is a decreasing sequence) so that
is a sum of positive termsand hence
the sequence (_{
2
} is monotonic increasing. Also,_{2)}

so for all _{2
1
} is bounded and by the Monotone Convergence Theorem
converges to some limit _{2)}_{1
}

**Subproof.**

Now consider . On the one hand we have_{
2+1
}

so ( is monotonic decreasing, and on the other
we have_{2+1)}

which is a sum of positive terms hence positive. Thus ( is
decreasing and bounded below by 0,
and hence by the Monotone Convergence Theorem converges to some _{2+1)}_{2
}

**Subproof.**

Finally we show _{1=
2
}
as _{2+1-2=2+1
0
}_{2+1)} is a subsequence
of the null sequence (_{
)
}. But by continuity of -
we also have
so _{2+1-2
2-
1
}_{2-
1=0
}
_{1=
2
}
converge to the same limit _{
}_{1=
2
}
as
_{
1=
2
}

Example.

The series
^{
+1
}
^{
+1
}
*not* converge. (One is the harmonic series;
the other can be proved divergent by comparison with the
harmonic series.)

We saw the series
^{
+1
}

Definition.

So for example, ^{
+1
}1*all* conditionally convergent series.
Conditionally convergent series are particularly badly behaved,
should be avoided if possible, and *always* need great care. The results
we have proved about this one *are* correct though:
according to the definitions we have given,
the sum of the series ^{
+1
}
*does* exist and *is* greater than a half. But a rearrangement of the
series can make the sum less than a half, or in fact, any number you
like. To avoid such bad

cases you need to look at
absolutely convergent series, which are
the subject of the next web page.

You have seen the definition of alternating series, the alternating series test, and some examples. You have also seen how many such series behave very badly with respect to rearrangement of their terms.