Absolute convergence and conditional convergence.

Definition 9.5.1   A series $ \underset{n=0}{\overset{+\infty }{\sum}} u_n$ is absolutely convergent if the series $ \underset{n=0}{\overset{+\infty }{\sum}} \vert u_n\vert$ is convergent.

Proposition 9.5.2   If a series is absolutely convergent, it is convergent.

Example 9.5.3   Consider the series $ \underset{n=1}{\overset{+\infty }{\sum}} \frac {\cos n}{n^3}$ . The corresponding series of absolute values is $ \underset{n=1}{\overset{+\infty }{\sum}} \left\vert \frac {\cos n}{n^3} \right\vert
=\underset{n=1}{\overset{+\infty }{\sum}} \frac { \vert \cos n \vert }{n^3}$ is convergent because for any $ n$ , its general term is less than or equal to $ \frac {1}{n^3}$ (v.s. 3.3).

Thus, the given series is absolutely convergent, and is convergent.

\begin{figure}\mbox{\subfigure{\epsfig{file=error.eps, height=1cm}} \qquad
The converse is not true\vert}
\end{figure}
For example, the series $ \underset{n=0}{\overset{+\infty }{\sum}} \frac {(-1)^n}{n}$ is convergent, but not absolutely convergent, because the series $ \underset{n=0}{\overset{+\infty }{\sum}} \frac 1n$ is divergent, by Prop. 3.1.

A convergent series which is convergent, but not absolutely convergent is conditionally convergent.

For example, the alternating harmonic series is convergent (v.s. 4.3), but not absolutely convergent (v.s. 1.9), thus it is conditionally convergent.

Theorem 9.5.4   Let $ \underset{n=0}{\overset{+\infty }{\sum}} u_n$ and $ \underset{n=0}{\overset{+\infty }{\sum}} v_n$ be two series, where the sequence $ (v_n)$ is a rearrangement of the sequence $ (u_n)$ .

If $ \underset{n=0}{\overset{+\infty }{\sum}} u_n$ is absolutely convergent, then $ \underset{n=0}{\overset{+\infty }{\sum}} v_n$ is absolutely convergent, and the two series have the same sum.

Remark 9.5.5   If the series $ \underset{n=0}{\overset{+\infty }{\sum}} u_n$ is conditionally convergent, we can reaarange its terms so that it will show any predecided behaviour: divergence or convergence (with any limit we want!). For example, consider the alternating harmonic sequence.

Noah Dana-Picard 2007-12-28