Partial Sums

Definition: Partial Sum

Let ${z_{n}}_{n \in D}$ be a complex sequence.

The $k$ -th partial sum of ${z_{n}}$ is the sum of its first $k$ numbers:
$j j \in D \leq k \sum a_{j}$

NOTATION

$s_{k}$

Complex Series

Given a complex sequence $(z_{n})_{n \in D}$ , it is apparent that the function which maps each $k \in D$ to the $k$ -th partial sum of $(z_{n})_{n \in D}$ is itself a complex sequence $(s_{n})_{n \in D}$ .

Definition: Complex Series

A complex series is the complex sequence of the partial sums of some other complex sequence.

NOTATION

If the sequence is $(z_{n})_{n \in D}$ , then we denote the sequence of its partial sums by $n \in D \sum z_{n}$ .

When $D = N_{0}$ or $D = N$ , we write $n = 0 \sum \infty z_{n}$ or $n = 1 \sum \infty z_{n}$ , respectively.

In the case, when $D = N ∖ {0, 1, \dots, p - 1}$ for some integer $p$ , we write $n = p \sum \infty z_{n}$ .

Convergence

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

Notation: Convergence of Complex Series

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

Definition: Absolute Convergence of Complex Series

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

Definition: Conditional Convergence

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

Theorem: Absolute Convergence $⟹$ Convergence

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

PROOF

TODO

Theorem: Test for Divergence

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

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

PROOF

TODO

Razon

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Complex Series

Partial Sums

Complex Series

Convergence

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