## Convergence and divergence.

Definition 7.1.3   The series of complex numbers is convergent if the sequence of partial sums is convergent. The limit is called the sum of the series. A non convergent series is called divergent.

Example 7.1.4

• The harmonic series is divergent.
• The alternating harmonic series is convergent.
• A geometric series is convergent if, and only if .
• The Riemann series is convergent if, and only if .

Definition 7.1.5   A series of complex numbers is absolutely convergent if the series 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