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       

• The harmonic series $\underset{k=1}{\overset{+\infty }{\sum}} \frac 1k$ is divergent.
• The alternating harmonic series $\underset{k=1}{\overset{+\infty }{\sum}} \frac {(-1)^{n+1}}{k}$ is convergent.
• A geometric series $\underset{k=0}{\overset{+\infty }{\sum}} q^k$ is convergent if, and only if $\vert q\vert<1$ .
• The Riemann $p-$ series $\underset{k=1}{\overset{+\infty }{\sum}} \frac {1}{n^p}$ is convergent if, and only if $p>1$ .

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.