Complex Analysis/Sequences and series

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:

$${\displaystyle \lim _{n\to \infty }a_{n}=a_{0}\quad$$ :\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 ${\displaystyle a_{n}\to a_{0}\in \mathbb {C} }$ resp. ${\displaystyle a_{n}{}_{\stackrel {\displaystyle \longrightarrow }{n\to \infty }}a_{0}\in \mathbb {C} }$ 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 ${\displaystyle \varepsilon }$ environment of ${\displaystyle a_{o}}$ (i.e. ${\textstyle a_{n}\in \,\,]a_{o}-\varepsilon ,a_{o}+\varepsilon [}$

Semantics

The meaning of the definition can be promised as follows:

$${\displaystyle \lim _{n\to \infty }a_{n}=a_{0}:\Leftrightarrow \forall \varepsilon >0\;\exists n_{\varepsilon }\in \mathbb {N} \;\forall n\geq n_{\varepsilon }:\;\left|a_{n}-a_{0}\right|<\varepsilon }$$.

For an arbitrarily small ${\displaystyle \varepsilon >0}$ and you can always find an index bound ${\textstyle n_{\varepsilon }\in \mathbb {N} }$ from distance to ${\displaystyle a_{o}}$ is smaller than ${\displaystyle \varepsilon }$ (i.e. ${\displaystyle |a_{m}-a_{o}|<\varepsilon }$

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 ${\displaystyle \mathbb {C} }$.

Learning Tasks

• Prove that the complex series ${\textstyle \sum _{k=1}^{\infty }(-1)^{k}\cdot {\frac {1}{k}}\cdot i}$ converges in ${\displaystyle \mathbb {C} }$, 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 ${\displaystyle \sum _{k=1}^{\infty }a_{k}}$ 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 ${\displaystyle \sum _{k=1}^{\infty }a_{k}}$ defines a complex number as a limit of the sequence of partial sums:

${\displaystyle \sum _{k=1}^{\infty }a_{k}=\lim _{n\to \infty }s_{n}}$

with ${\displaystyle s_{n}:=\sum _{k=1}^{n}|a_{k}|.}$

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.:

$${\displaystyle \sum _{n=1}^{\infty }a_{n}=s_{0}:\Leftrightarrow \forall \varepsilon >0\;\exists n_{\varepsilon }\in \mathbb {N} \;\forall n\geq n_{\varepsilon }:\;\left|s_{n}-s_{0}\right|<\varepsilon .}$$

This applies ${\textstyle \left|s_{n}-s_{0}\right|=\left|\sum _{k=1}^{n}a_{k}-s_{0}\right|<\varepsilon }$.

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

Wiki2Reveal

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