6.1 Sequences and series of complex numbers

Complex sequences and series are fundamental to understanding analytic functions. They extend real number concepts to the complex plane, allowing us to explore convergence and divergence in two dimensions. These tools are crucial for representing functions as power series.

Convergence tests for complex series build on real series techniques. The comparison, ratio, and root tests help determine if a series converges absolutely or conditionally. Geometric and telescoping series provide important examples of convergent complex series with known sums.

Sequences and series in the complex domain

Complex sequences and series definitions

• A complex sequence maps natural numbers to complex numbers, denoted as $\{z_n\}$ where $z_n$ represents the nth term
• The limit $L$ of a complex sequence $\{z_n\}$ satisfies $|z_n - L| < \epsilon$ for all $n > N$, given any real number $\epsilon > 0$ and some natural number $N$
• A complex series, denoted as $\sum z_n$, adds the terms of a complex sequence
• The nth partial sum of a complex series $\sum z_n$, denoted as $s_n$, equals $z_1 + z_2 + ... + z_n$
• A complex series $\sum z_n$ converges when its partial sum sequence $\{s_n\}$ approaches a complex number $S$, called the series sum

Convergence properties of complex sequences and series

• A complex sequence $\{z_n\}$ converges if and only if its real part $\{Re(z_n)\}$ and imaginary part $\{Im(z_n)\}$ sequences converge
• The absolute value sequence $\{|z_n|\}$ of a convergent complex sequence $\{z_n\}$ with limit $L$ converges to $|L|$
• A complex series $\sum z_n$ converges if and only if its real part series $\sum Re(z_n)$ and imaginary part series $\sum Im(z_n)$ converge
• The term sequence $\{z_n\}$ of a convergent complex series $\sum z_n$ must approach 0
• The Cauchy criterion states that $\sum z_n$ converges if and only if for every $\epsilon > 0$, there exists $N$ such that $|z_{n+1} + ... + z_{n+p}| < \epsilon$ for all $n > N$ and all $p \geq 1$

Convergence of complex sequences and series

Limits and convergence of complex sequences

• A complex sequence $\{z_n\}$ converges to a limit $L$ if for any $\epsilon > 0$, there exists an $N$ such that $|z_n - L| < \epsilon$ for all $n > N$
• Convergence of complex sequences requires both the real and imaginary parts to converge as real sequences ($\{Re(z_n)\}$ and $\{Im(z_n)\}$)
• If a complex sequence $\{z_n\}$ converges to $L$, then the sequence of its absolute values $\{|z_n|\}$ converges to $|L|$
• Examples of convergent complex sequences include $\{1/n + i/n^2\}$ (converges to 0) and $\{e^{i\theta}/n\}$ (converges to 0 for any real $\theta$)

Convergence of complex series

• A complex series $\sum z_n$ converges if and only if both its real part series $\sum Re(z_n)$ and imaginary part series $\sum Im(z_n)$ converge as real series
• For a complex series $\sum z_n$ to converge, the sequence of its terms $\{z_n\}$ must converge to 0
• The Cauchy criterion for complex series states that $\sum z_n$ converges if and only if for every $\epsilon > 0$, there exists an $N$ such that $|z_{n+1} + ... + z_{n+p}| < \epsilon$ for all $n > N$ and all $p \geq 1$
• Examples of convergent complex series include $\sum \frac{1}{n^2+in}$ and $\sum \frac{(-1)^n(1+i)}{n}$

Tests for convergence of complex series

Comparison and absolute convergence tests

• The comparison test states that if $|z_n| \leq a_n$ for all $n$, where $\sum a_n$ is a convergent real series, then $\sum z_n$ converges
• Absolute convergence of a complex series $\sum z_n$ implies its convergence, but a series may converge conditionally without absolute convergence
• Examples of applying the comparison test include comparing $\sum \frac{i^n}{n^2}$ with $\sum \frac{1}{n^2}$ (convergent) and $\sum \frac{1+i}{n}$ with $\sum \frac{1}{n}$ (divergent)

Ratio and root tests for complex series

• The ratio test states that if $\lim_{n\to\infty} |\frac{z_{n+1}}{z_n}| = L$, then $\sum z_n$ converges absolutely for $L < 1$, diverges for $L > 1$, and is inconclusive for $L = 1$
• The root test states that if $\lim_{n\to\infty} (|z_n|)^{1/n} = L$, then $\sum z_n$ converges absolutely for $L < 1$, diverges for $L > 1$, and is inconclusive for $L = 1$
• Examples of applying the ratio test include $\sum \frac{(1+i)^n}{n!}$ (converges) and $\sum n!(1-i)^n$ (diverges)
• Examples of applying the root test include $\sum \frac{(1+i)^n}{n^n}$ (converges) and $\sum \frac{n^n(1-i)^n}{n!}$ (diverges)

Sum of convergent geometric and telescoping series

Complex geometric series and their sums

• A complex geometric series has the form $\sum z^n = a + az + az^2 + ...$, where $a$ and $z$ are complex numbers
• A complex geometric series converges if and only if $|z| < 1$, and its sum equals $\frac{a}{1-z}$
• The sum of a finite geometric series with $n$ terms and common ratio $z$ is given by $s_n = \frac{a(1-z^n)}{1-z}$, where $a$ is the first term
• Examples of complex geometric series include $\sum (\frac{1+i}{2})^n$ (converges to $\frac{2}{1-i}$) and $\sum (1-i)^n$ (diverges)

Telescoping series and their sums

• A telescoping series has most terms cancel out, leaving only a finite number of terms contributing to the sum
• The partial sums of a telescoping series can be written as $s_n = (b_1 - c_1) + (b_2 - c_2) + ... + (b_n - c_n)$, where each $c_i$ cancels with the $b_{i+1}$ term, except for the first and last terms
• The sum of a convergent telescoping series equals the limit of its partial sums, found by evaluating the remaining terms after cancellation
• Examples of telescoping series include $\sum \frac{1}{n(n+1)}$ (converges to 1) and $\sum \frac{i^n}{n} - \frac{i^{n+1}}{n+1}$ (converges to $1-i$)