If a series has some positive and some negative terms, there are
a couple of things that one might ask. The first is
1) does the series converge?
Another question, the motivation for which is less
obvious, is
2) does the series converge if we take
the absolute values of its terms?
If the first answer is yes, the second can be yes or no. It
turns out that if this second answer is yes, the series behaves
much like a finite sum, i.e. it behaves well.
Definition:
Let $\sum a_n$ be a series.
If the series $ {\sum \left|a_n\right|}$ converges, then
we say that $\sum a_n$ is absolutely convergent.
If $\sum a_n$ converges but $\sum \left|a_n\right|$
doesn't, then we say that $\sum a_n$ is conditionally
convergent.
Example: Consider the alternating harmonic series
$$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac12 + \frac13 -
\frac14+\cdots.$$It converges (we saw this previously by
using the AST). The series with the absolute values of its
terms, which is the harmonic series $\sum \frac{1}{n}$, diverges
($p$-series with $p\le 1$). Since the
series converges, but not in absolute value, we say it is
conditionally convergent.
One fact, said in two ways
If $\sum \left|a_n\right|$ converges, then $\sum a_n$
converges. (Absolutely convergent $\Longrightarrow$ convergent.)
If $\sum a_n$ does not converge, then $\sum\left|a_n\right|$
will not converge. (Divergent $\Longrightarrow$ not absolutely
convergent.)
Example: Consider the alternating $p$-series, with $p=2$,
$$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^2} = 1 - \frac14 + \frac19 -
\frac1{16}+\cdots.$$Since the series with the absolute values of the
terms of our series, $\sum\frac{1}{n^2}$, is a convergent
$p$-series, our series is absolutely
convergent. By the fact above, this means it is also convergent.It is not conditionally convergent.
Be careful with these terms
Conditional convergence of a
series means it is convergent but not
absolutely convergent.
If we are told that a series is convergent,
we do not know a priori whether it is conditionally convergent or absolutely
convergent. It is one
or the other, but not both. Every series is either
divergent, conditionally convergent, or absolutely convergent, but
it is only one of these things.
Justification of the fact above, and some examples, are discussed in
the video.