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 ${\textstyle n}$ is called ${\textstyle n}$-tes member or ${\textstyle n}$-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 ${\textstyle I}$ and a basic space ${\textstyle G}$ from which the components are selected, then one can understand a sequence ${\textstyle a}$ sequences are usually listed ${\textstyle (a_{n})_{n\in I}\in G^{I}}$ (e.g. ${\textstyle I=\mathbb {N} )}$

Examples of sequences

• ${\textstyle (a_{1},a_{2},a_{3})\in \mathbb {R} ^{3}}$ finite real number sequence or ${\textstyle n}$LINK_INT_1730533169719_4___
• ${\textstyle (a_{n})_{n\in \mathbb {N} }=\left({\frac {1}{n}}\right)_{n\in \mathbb {N} }\in \mathbb {R} ^{\mathbb {N} }}$ is a real number sequence.
• ${\textstyle (a_{n})_{n\in \mathbb {N} }=\left({\frac {5+n}{n}}+{\frac {5+3n^{2}}{4n^{2}}}i\right)_{n\in \mathbb {N} }\in \mathbb {C} ^{\mathbb {N} }}$ is a complex sequence of numbers.

In the sequencesden we consider infinite sequences with the index quantity ${\textstyle I=\mathbb {N} }$ 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 ${\textstyle (a_{n})_{n\in \mathbb {N} }}$ against a limit value ${\textstyle a_{0}\in G}$. The distance of the sequence elements to the limit value runs against 0.

Notation

For the limit value ${\textstyle a_{0}\in \mathbb {C} }$ of a sequence ${\textstyle (a_{n})_{n\in \mathbb {N} }\in \mathbb {C} ^{\mathbb {N} }}$ there is a separate symbol, you write:

${\textstyle \lim _{n\to \infty }a_{n}=a_{0}}$ with ${\textstyle n\in \mathbb {N} }$.

Definition sequence convergence

Be ${\textstyle (a_{n})_{n\in \mathbb {N} }\in \mathbb {C} ^{\mathbb {N} }}$ a complex number sequence and ${\textstyle a_{0}\in \mathbb {C} }$. The convergence of ${\textstyle (a_{n})_{n\in \mathbb {N} }}$ against ${\textstyle a_{0}}$ 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

• ${\textstyle a_{n}\to a_{0}}$ for ${\textstyle n\to \infty }$ 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

Epsilon Environment and convergence

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

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 ${\textstyle n_{\varepsilon }\in \mathbb {N} }$ from distance to $a_{o}$ is smaller than $\varepsilon$ (i.e. $|a_{m}-a_{o}|<\varepsilon$

Series

A series ${\textstyle \displaystyle \sum _{k=1}^{\infty }a_{k}}$ is a special sequence ${\textstyle (s_{n})_{n\in \mathbb {N} }}$, 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 ${\textstyle \displaystyle \sum _{k=1}^{\infty }a_{k}}$ converges if the associated sequence of the partial sums ${\textstyle (s_{n})_{n\in \mathbb {N} }:=\displaystyle \left(\sum _{k=1}^{n}a_{k}\right)_{n\in \mathbb {N} }}$ converges as sequence in vector space.

Absolute convergence

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

Learning Tasks

• Prove that the complex series ${\textstyle \sum _{k=1}^{\infty }(-1)^{k}\cdot {\frac {1}{k}}\cdot i}$ converges in $\mathbb {C}$, but does not converge absolutely. Use a theorem and your knowledge from calculus.
• Check the following series ${\textstyle \sum _{k=1}^{\infty }(i)^{k}\cdot 2^{-k+2}}$ 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 ${\textstyle (s_{n})_{n\in \mathbb {N} }}$. 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 ${\textstyle s_{0}\in \mathbb {C} }$ of a series ${\textstyle \sum _{n=1}^{\infty }a_{n}}$ with ${\textstyle (a_{n})_{n\in \mathbb {N} }\in \mathbb {C} ^{\mathbb {N} }}$, you write:

${\textstyle \sum _{n=1}^{\infty }a_{n}=s_{0}}$

Convergence of Series

Be ${\textstyle (a_{n})_{n\in \mathbb {N} }\in \mathbb {C} ^{\mathbb {N} }}$ a complex number sequence and ${\textstyle s_{0}\in \mathbb {C} }$. The convergence of the series refers to the convergence of the partial sums ${\textstyle s_{n}=\sum _{k=1}^{n}a_{k}}$ against ${\textstyle s_{0}}$, 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 ${\textstyle \left|s_{n}-s_{0}\right|=\left|\sum _{k=1}^{n}a_{k}-s_{0}\right|<\varepsilon }$.

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 ${\textstyle V}$

• an additive operation to calculate vectors for the partial sums ${\textstyle s_{n}:=\sum _{k=1}^{n}a_{n}}$
• a topology to be able to investigate the convergence of the sequence of partial sums ${\textstyle (s_{n})_{n\in \mathbb {N} }}$.

Task - series convergence in a topological vector space

Let ${\textstyle a_{n}:={\begin{pmatrix}2^{-n}\\3^{-n}\\4^{-n}\end{pmatrix}}\in \mathbb {R} ^{3}}$. Calculate the limit of the series ${\textstyle g:=\sum _{k=0}^{\infty }a_{n}\in \mathbb {R} ^{3}}$ 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 ${\textstyle \mathbb {C} }$. Transfer the definition convergence sequences or series to topological vector spaces ${\textstyle (V,\|\cdot \|)}$. What are the similarities and differences?

Topological algebra - potency series

Let ${\textstyle A}$ be a given topological algebra, then denote the algebra of power series by ${\textstyle {\overline {A[t]}}}$

• with a topology on the algebra ${\textstyle {\overline {A[t]}}}$ is the topological closure on the algebra ${\textstyle A[t]}$ 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 ${\textstyle A[t]}$ and the algebra of power series ${\textstyle {\overline {A[t]}}}$.

Example - Polynomial Algebra and Convex Combinations

In the vector space ${\textstyle n\in \mathbb {N} }$ in the vector space ${\textstyle \mathbb {R} ^{3}}$, 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

See also

• sequences in der Mathematik - Wikipedia
• Gaussian number and coordinate systems
• convex combination
• topological vector space
• topological algebra
• norms, metrics, topology

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