MATH529: L22 Series in $ℂ$

Overview

• Sequences and series in $ℂ$

• Taylor series

Sequence convergence

• Sequence $\left\{z_n\right\}$ converges to $L\in ℂ$ iff $\mathrm{Re}{z}_n\to \mathrm{Re}L$ and $\mathrm{Im}{z}_n\to \mathrm{Im}L$

• $\forall \epsilon >0,\exists n_{\epsilon }\in \mathbb{N},s.t.n>n_{\epsilon },|z_n-L|<\epsilon$, $z_n=x_n+i{y}_n$, $L=M+iN$

  ${\displaystyle |z_n-L|={[{(x_n-M)}^2+{(y_n-N)}^2]}^{1/2}}$

• Examples:

  • $z_n=5i^n$, Convergent? No

  • $z_n=1+e^{n\pi i}$, Convergent? No

  • $z_n=\frac{n+i^n}{\sqrt{n}}$, Convergent? No

  • $z_n=\frac{i^n+1}{n+i}$, Convergent? Yes

• Motivation for sequences: ensure closure of $ℝ$, $e={lim}_{n\to \infty }{(1+\frac{1}{n})}^n$, $x_n={(1+\frac{1}{n})}^n\to e$

Series in $ℂ$

• The series ${\sum }_{k=1}^{\infty }{z}_k$ is convergent if the sequence of partial sums $S_n$converges

  ${\displaystyle S_n=\sum _{k=1}^n{z}_k,{\left\{S_n\right\}}_{n\in \mathbb{N}}}$

• The geometric series ${\sum }_{k=1}^{\infty }{z}^{k-1}$ converges to $1/(1-z)$ when $\left|z\right|<1$

  ${\displaystyle S_n=\sum _{k=1}^n{z}^{k-1}=1+z+\cdots +z^{n-1}=\frac{1-z^n}{1-z}\to \frac{1}{1-z}}$

• Note that

  ${\displaystyle \frac{1}{1-z}=1+z+\cdots +z^{n-1}+\frac{z^n}{1-z}}$

  of interest in applications of Cauchy's integral formula $f\left(z_0\right)=\frac{1}{2\pi i}{\oint }_C\frac{f\left(z\right)}{z-z_0}dz$

$ℂ$ Series convergence criteria

• If ${\sum }_{k=1}^{\infty }{z}_k$ converges then ${lim}_{n\to \infty }{z}_n=0$

• If ${lim}_{n\to \infty }{z}_n\ne 0$ then ${\sum }_{k=1}^{\infty }{z}_k$ diverges

• ${\sum }_{k=1}^{\infty }{z}_k$ is absolutely convergent if ${\sum }_{k=1}^{\infty }\left|z_k\right|$ is convergent

• Ratio test: series ${\sum }_{k=1}^{\infty }{z}_k$, with terms such that ${lim}_{n\to \infty }|z_{n+1}/z_n|=L$:

  1. if $L<1$ series is absolutely convergent

  2. if $L>1$ (including $L=\infty$) series is divergent

  3. if $L=1$ is inconclusive

• Root test: series ${\sum }_{k=1}^{\infty }{z}_k$, with terms such that ${lim}_{n\to \infty }{\left|z_n\right|}^{1/n}=L$:

  1. if $L<1$ series is absolutely convergent

  2. if $L>1$ (including $L=\infty$) series is divergent

  3. if $L=1$ is inconclusive

Power series

• As presaged by Cauchy's formula's power series are of particular interest

  ${\displaystyle \sum _{k=0}^{\infty }{a}_k{(z-z_0)}^k}$

• $z_0$ is the center of the series

• Ratio test on ${\sum }_{k=1}^{\infty }{z}^{k+1}/k$, absolutely convergent for $\left|z\right|<1$

  ${\displaystyle L={lim}_{n\to \infty }\frac{|z^{n+2}/(n+1)|}{|z^{n+1}/n|}={lim}_{n\to \infty }\frac{n}{n+1}\left|z\right|=\left|z\right|}$

• Circle of convergence: series ${\sum }_{k=0}^{\infty }{a}_k{(z-z_0)}^k$, $L={lim}_{n\to \infty }|a_{n+1}/a_n|$

  1. if $L\ne 0$ series has radius of convergence $R=1/L$, $|z-z_0|<R$

  2. if $L=0$ series converges everywhere $R=\infty$

  3. if $L=\infty$ radius of convergence is $R=0$

Taylor series

• ${\sum }_{k=0}^{\infty }{a}_k{(z-z_0)}^k$ represents a continuous function $f$ for $|z-z_0|<R$, $R\ne 0$

• Term-by-term integration is possible for any contour $C$ within $|z-z_0|<R\ne 0$

• Term-by-term differentiation is possible within $|z-z_0|<R\ne 0$

• $f\left(z\right)={\sum }_{k=0}^{\infty }{a}_k{(z-z_0)}^k$ is analytic for $|z-z_0|<R$, $R\ne 0$

  ${\displaystyle f^{\text{'}}\left(z\right)=\sum _{k=1}^{\infty }k{a}_k{(z-z_0)}^{k-1},f^{\text{'}}{}^{\text{'}}\left(z\right)=\sum _{k=2}^{\infty }k(k-1)a_k{(z-z_0)}^{k-2}}$ ${\displaystyle f\left(z_0\right)=0!a_0,f^{\text{'}}\left(z_0\right)=1!a_1,f^{\text{'}\text{'}}\left(z_0\right)=2!a_2,\dots }$

• Taylor's theorem $f:D\to ℂ$, $f$ analytic in $D$, $z_0\in D\subset ℂ$

  ${\displaystyle f\left(z\right)=\sum _{k=0}^{\infty }\frac{f^{\left(k\right)}\left(z_0\right)}{k!}{(z-z_0)}^k}$