# Complex Analysis/Sequences and series
## Introduction

In the Mathematics a listing (indexed family) of finally or infinitely many continuously numbered objects (for example numbers). The same object can also occur several times in a sequence. The object with the index n{\textstyle n} is called n{\textstyle n}-tes member or n{\textstyle n}-te component of the sequence. Endual and infinite sequences can be found in all areas of mathematics. With infinite sequences whose links are real numbers, the Analysis is concerned.

## sequences as images

If an index quantity I{\textstyle I} and a basic space G{\textstyle G} from which the components are selected, then one can understand a sequence a{\textstyle a} sequences are usually listed (a$_n$)$_{n∈I}$∈G$^I${\textstyle (a_{n})_{n\in I}\in G^{I}} (e.g. I=N){\textstyle I=\mathbb {N} )}

## Examples of sequences

* (a$_1$, a$_2$, a$_3$)∈R$^3${\textstyle (a_{1}, a_{2}, a_{3})\in \mathbb {R} ^{3}} finite real number sequence or n{\textstyle n}LINK_INT_1730533169719_4___
* (a$_n$)$_{n∈N}$=(1/n)$_{n∈N}$∈R$^N${\textstyle (a_{n})_{n\in \mathbb {N} }=\left({\frac {1}{n}}\right)_{n\in \mathbb {N} }\in \mathbb {R} ^{\mathbb {N} }} is a real number sequence.
* (a$_n$)$_{n∈N}$=( (5+n)/n+(5+3n$^2$)/4n$^2$ i)$_{n∈N}$∈C$^N${\textstyle (a_{n})_{n\in \mathbb {N} }=\left({\frac {5+n}{n}}+{\frac {5+3n^{2}}{4n^{2}}}i\right)_{n\in \mathbb {N} }\in \mathbb {C} ^{\mathbb {N} }} is a complex sequence of numbers.
In the sequencesden we consider infinite sequences with the index quantity I=N{\textstyle I=\mathbb {N} } in the complex numbers.

## Convergence of sequences

If distances with a metric or length of vectors with a standard can be measured on a basic space, the sequence convergence can be defined. For convergent sequences (a$_n$)$_{n∈N}${\textstyle (a_{n})_{n\in \mathbb {N} }} against a limit value a$_0$∈G{\textstyle a_{0}\in G}. The distance of the sequence elements to the limit value runs against 0.

### Notation

For the limit value a$_0$∈C{\textstyle a_{0}\in \mathbb {C} } of a sequence (a$_n$)$_{n∈N}$∈C$^N${\textstyle (a_{n})_{n\in \mathbb {N} }\in \mathbb {C} ^{\mathbb {N} }} there is a separate symbol, you write:
    lim$_{n→∞}$a$_n$=a$_0${\textstyle \lim _{n\to \infty }a_{n}=a_{0}} with n∈N{\textstyle n\in \mathbb {N} }.

### Definition sequence convergence

Be (a$_n$)$_{n∈N}$∈C$^N${\textstyle (a_{n})_{n\in \mathbb {N} }\in \mathbb {C} ^{\mathbb {N} }} a complex number sequence and a$_0$∈C{\textstyle a_{0}\in \mathbb {C} }. The convergence of (a$_n$)$_{n∈N}${\textstyle (a_{n})_{n\in \mathbb {N} }} against a$_0${\textstyle a_{0}} is then defined as follows:
    $\lim _{n\to \infty }a_{n}=a_{0}\quad :\Longleftrightarrow \quad \forall \varepsilon >0\;\exists n_{\varepsilon }\in \mathbb {N} \;\forall n\geq n_{\varepsilon }:\;\left|a_{n}-a_{0}\right|<\varepsilon$.

### Notation

In addition to the notation above, we can denote
* a$_n$→a$_0${\textstyle a_{n}\to a_{0}} for n→∞{\textstyle n\to \infty } or
* just $a_{n}\to a_{0}\in \mathbb {C}$ resp. $a_{n}_{\stackrel {\displaystyle \longrightarrow }{n\to \infty }}a_{0}\in \mathbb {C}$ can be used.

### Visualization of convergence

For all ε>0{\textstyle \varepsilon >0} there is an index bound n$_ε${\textstyle n_{\varepsilon }} from which all components a$_n${\textstyle a_{n}} of the sequence are an element of $\varepsilon$ environment of $a_{o}$ (i.e. a$_n$∈]a$_o$−ε, a$_o$+ε[{\textstyle a_{n}\in \, \, ]a_{o}-\varepsilon , a_{o}+\varepsilon

### Semantics

The meaning of the definition can be promised as follows:
    $\lim _{n\to \infty }a_{n}=a_{0}:\Leftrightarrow \forall \varepsilon >0\;\exists n_{\varepsilon }\in \mathbb {N} \;\forall n\geq n_{\varepsilon }:\;\left|a_{n}-a_{0}\right|<\varepsilon$.
For an arbitrarily small $\varepsilon >0$ and you can always find an index bound n$_ε$∈N{\textstyle n_{\varepsilon }\in \mathbb {N} } from distance to $a_{o}$ is smaller than $\varepsilon$ (i.e. $|a_{m}-a_{o}|<\varepsilon$

## Series

A series ∑$_{k=1}$$^∞$a$_k${\textstyle \displaystyle \sum _{k=1}^{\infty }a_{k}} is a special sequence (s$_n$)$_{n∈N}${\textstyle (s_{n})_{n\in \mathbb {N} }}, which is generated from a given sequence $(a_{n})_{n\in \mathbb {N} }$ by the sequence of the partial sums $(s_{n})_{n\in \mathbb {N} }:=\left(\sum _{k=1}^{n}a_{k}\right)_{n\in \mathbb {N} }$.

### Convergence of series

If the underlying vector space has a topology (e.g. generated by a metric space or a norm then the convergence of a series can be expression in analogue way ∑$_{k=1}$$^∞$a$_k${\textstyle \displaystyle \sum _{k=1}^{\infty }a_{k}} converges if the associated sequence of the partial sums (s$_n$)$_{n∈N}$:=(∑$_{k=1}$$^n$a$_k$)$_{n∈N}${\textstyle (s_{n})_{n\in \mathbb {N} }:=\displaystyle \left(\sum _{k=1}^{n}a_{k}\right)_{n\in \mathbb {N} }} converges as sequence in vector space.

### Absolute convergence

A series ∑$_{k=1}$$^∞$a$_k${\textstyle \sum _{k=1}^{\infty }a_{k}} is absolutely convergent when the series ∑$_{k=1}$$^∞$|a$_k$|{\textstyle \sum _{k=1}^{\infty }|a_{k}|}converges, i.e. the sequence of the partial sums (s$_n$)$_{n∈N}$:=(∑$_{k=1}$$^n$|a$_k$|)$_{n∈N}${\textstyle (s_{n})_{n\in \mathbb {N} }:=\left(\sum _{k=1}^{n}|a_{k}|\right)_{n\in \mathbb {N} }} converges in $\mathbb {C}$.

### Learning Tasks

* Prove that the complex series ∑$_{k=1}$$^∞$(−1)$^k$⋅ 1/k ⋅i{\textstyle \sum _{k=1}^{\infty }(-1)^{k}\cdot {\frac {1}{k}}\cdot i} converges in $\mathbb {C}$, but does not converge absolutely. Use a theorem and your knowledge from calculus.
* Check the following series ∑$_{k=1}$$^∞$(i)$^k$⋅2$^{−k+2}${\textstyle \sum _{k=1}^{\infty }(i)^{k}\cdot 2^{-k+2}} for convergence. Calculate the first components of the partial sum as plot those numbers in the Gaussian number in a coordinate system. Explain why the sequence of the partial sums have these geometric properties. Calculate the limit of the series if it exists.

### Series as mathematical object

The term $\sum _{k=1}^{\infty }a_{k}$ of a series does not denote sequence of partial sums (s$_n$)$_{n∈N}${\textstyle (s_{n})_{n\in \mathbb {N} }}. In the case of a convergence the expression $\sum _{k=1}^{\infty }a_{k}$ defines a complex number as a limit of the sequence of partial sums:
    $\sum _{k=1}^{\infty }a_{k}=\lim _{n\to \infty }s_{n}$
with $s_{n}:=\sum _{k=1}^{n}|a_{k}|.$

### Notation

For the limit value s$_0$∈C{\textstyle s_{0}\in \mathbb {C} } of a series ∑$_{n=1}$$^∞$a$_n${\textstyle \sum _{n=1}^{\infty }a_{n}} with (a$_n$)$_{n∈N}$∈C$^N${\textstyle (a_{n})_{n\in \mathbb {N} }\in \mathbb {C} ^{\mathbb {N} }}, you write:
    ∑$_{n=1}$$^∞$a$_n$=s$_0${\textstyle \sum _{n=1}^{\infty }a_{n}=s_{0}}

### Convergence of Series

Be (a$_n$)$_{n∈N}$∈C$^N${\textstyle (a_{n})_{n\in \mathbb {N} }\in \mathbb {C} ^{\mathbb {N} }} a complex number sequence and s$_0$∈C{\textstyle s_{0}\in \mathbb {C} }. The convergence of the series refers to the convergence of the partial sums s$_n$=∑$_{k=1}$$^n$a$_k${\textstyle s_{n}=\sum _{k=1}^{n}a_{k}} against s$_0${\textstyle s_{0}}, i.e.:
    $\sum _{n=1}^{\infty }a_{n}=s_{0}:\Leftrightarrow \forall \varepsilon >0\;\exists n_{\varepsilon }\in \mathbb {N} \;\forall n\geq n_{\varepsilon }:\;\left|s_{n}-s_{0}\right|<\varepsilon .$
This applies |s$_n$−s$_0$|=|∑$_{k=1}$$^n$a$_k$−s$_0$|<ε{\textstyle \left|s_{n}-s_{0}\right|=\left|\sum _{k=1}^{n}a_{k}-s_{0}\right|<\varepsilon }.

## Convergence of series with coefficients from an algebra

The consideration of series convergence is a special case of the convergence of power series in a topological vector space or on a topological algebra.

### Topological vector space

The topological vector space V{\textstyle V}
* an additive operation to calculate vectors for the partial sums s$_n$:=∑$_{k=1}$$^n$a$_n${\textstyle s_{n}:=\sum _{k=1}^{n}a_{n}}
* a topology to be able to investigate the convergence of the sequence of partial sums (s$_n$)$_{n∈N}${\textstyle (s_{n})_{n\in \mathbb {N} }}.

### Task - series convergence in a topological vector space

Let a$_n$:=(2$^{−n}$3$^{−n}$4$^{−n}$)∈R$^3${\textstyle a_{n}:={\begin{pmatrix}2^{-n}\\3^{-n}\\4^{-n}\end{pmatrix}}\in \mathbb {R} ^{3}}. Calculate the limit of the series g:=∑$_{k=0}$$^∞$a$_n$∈R$^3${\textstyle g:=\sum _{k=0}^{\infty }a_{n}\in \mathbb {R} ^{3}} as an introductory example with the tools of the analysis! Explain the similarities and differences of the series convergence in topological vector spaces and the special case C{\textstyle \mathbb {C} }. Transfer the definition convergence sequences or series to topological vector spaces (V, ‖⋅‖){\textstyle (V, \|\cdot \|)}. What are the similarities and differences?

### Topological algebra - potency series

Let A{\textstyle A} be a given topological algebra, then denote the algebra of power series by A[t]$^¯${\textstyle {\overline {A[t]}}}
* with a topology on the algebra A[t]$^¯${\textstyle {\overline {A[t]}}} is the topological closure on the algebra A[t]{\textstyle A[t]} of polynomial with coefficients
* an topology is generated by summation of the weighted vector lengths of the coefficients,
* additive and multiplicative operation can also be defined similar to the agebra polynomials A[t]{\textstyle A[t]} and the algebra of power series A[t]$^¯${\textstyle {\overline {A[t]}}}.

### Example - Polynomial Algebra and Convex Combinations

In the vector space n∈N{\textstyle n\in \mathbb {N} } in the vector space R$^3${\textstyle \mathbb {R} ^{3}}, these polynomials of the order 3 represent With CAS4Wiki, the track of the three-dimensional convex combination of the order 3 can be plotted. CAS4Wiki Commands

## Literature

* Bourbaki: Éléments de mathématique. Theory of the Ensemble II/ III, Paris 1970
* Harro Heuser: Lehrbuch der Analysis, Part 1, Teubner Verlag, Stuttgart
References/>

## Web links

* Online encyclopedia of the number sequences
* [http://www.mathematik-wissen.de/dateness.htm
* sequences and Tupel in the Encyclopaedia of Mathematics, Springer, Edited by Michiel Hazewinkel