Convergence and divergence.

Definition 7.1.3   The series $ \underset{k=0}{\overset{+\infty }{\sum}} u_k$ of complex numbers is convergent if the sequence of partial sums $ S_n= \underset{k=0}{\overset{n}{\sum}} u_k$ is convergent. The limit is called the sum of the series. A non convergent series is called divergent.

Example 7.1.4       

Definition 7.1.5   A series $ \underset{n=0}{\overset{+\infty }{\sum}} u_n$ of complex numbers is absolutely convergent if the series $ \underset{n=0}{\overset{+\infty }{\sum}} \vert u_n\vert$ is convergent.

Proposition 7.1.6   If a series is absolutely convergent, it is convergent.

The converse is not true: for example, the alternating harmonic series is convergent, but not absolutely convergent, thus it is conditionally convergent.

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

Noah Dana-Picard 2007-12-24