Convergence

Since a complex series $n \in D \sum a_{n}$ is just a complex sequence, the definitions and terminology for convergence of series is the same as those used for convergence of sequences. What is different, however, is the notation used.

Notation: Convergence of Complex Series

If $n \in D \sum a_{n}$ converges to $L$ , we write
$n \in D \sum a_{n} = L$
Instead of calling $L$ the limit of $n \in D \sum a_{n}$ , we usually say it is its value.

Absolute Convergence

Definition: Absolute Convergence of Complex Series

A complex series $n \in D \sum a_{n}$ converges absolutely to $L$ iff the real series $n \in D \sum ∣ a_{n} ∣$ converges to $L$ .

Theorem: Absolute Convergence $⟹$ Convergence

If a complex series is absolutely convergent towards $L \in C$ , then it is also convergent towards $L$ .

PROOF

TODO

Conditional Convergence

Definition: Conditional Convergence

A complex series is conditionally convergent iff it is convergent but not absolutely convergent.

Criteria for Convergence and Divergence

Theorem: Test for Divergence

Let $n \in D \sum a_{n}$ be a complex series.

If the sequence $(a_{n})_{n \in D}$ diverges or its limit is not zero, then $n \in D \sum a_{n}$ diverges.

PROOF

TODO

Razon

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Convergence of Complex Series

Convergence

Absolute Convergence

Conditional Convergence

Criteria for Convergence and Divergence

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