The first part of this lecture was finishing the Poisson integral formula and is written in the previous document.
§60 (8 Ed §55)
Compare this to the situation in \mathbb R. Formally, a sequence is a function f : \mathbb N \to \mathbb C (or \mathbb N _0\to \mathbb C), n \mapsto z_n, written as \{z_n\}.
Definition (Limit). We say \lim_{n \to \infty}z_n = z or “\{z_n\} converges to z” if and only if given \epsilon > 0, there exists N \in \mathbb N such that n > N implies |z_n-z|<\epsilon. As in \mathbb R, this definition does not help us find a limit.
Definition (Series). Formally, \sum_{n=0}^\infty z_n for z_n \in \mathbb C converges as a series if and only if the associated sequence of partial sums \{s_n\} converges as a sequence, where s_n = \sum_{k=0}^n z_k.
A typical question we might ask is does \sum z_n converge?
An easy test that a series does not converge is the n-th term test, if \sum z_n converges, then z_n \to 0 (converse does not hold). Once we know it converges, we know that \sum z_n is just a complex number.
Remark: A sequence \{z_n\} is bounded if there exists M such that |z_n|<M for all n.
Convergent implies that a sequence is bounded (converse does not hold, see \{(-1)^n\}).
Definition (Absolute convergence). We say that \sum z_n converges absolutely if and only if \sum |z_n| converges. Absolute convergence implies convergence (converse does not hold, see \sum (-1)^n/n).
Definition (Remainder). Given \sum_{n=0}^\infty z_n, set s_n=\sum_{k=0}^n z_k as the partial sums. Then, let \rho_n = \sum_{k=n+1}^\infty z_k as the tail or remainder.
Theorem. s_n \to s if and only if \rho_n \to 0.
Example: We claim that \sum_{n=0}^\infty z_n = 1/(1-z)=S for |z| < 1.
Proof. \begin{aligned} s_n &= 1 + z + \cdots + z^n \\ zs_N &= z + z^2 + \cdots + z^{n+1} \\ \implies (1-z)s_n &= 1 - z^{n-1} \\ s_n &= \frac{1-z^{n+1}}{1-z}\\ \implies \rho_n=s-s_n&= \frac {z^{n+1}}{1-z} \end{aligned} Since |z|<1, |\rho_n| \to 0 as n \to \infty which implies that s_n \to s. \square
Remark: As in \mathbb R, we can do simple operations on convergent series:
Definition (Power series). A power series centred at z_0 is a series of the form \sum_{n=0}^\infty a_n(z-z_0)^n. This series has a radius of convergence. R. That is, it converges absolutely within this radius, diverges outside, and may converge or diverge on the boundary. (These can be checked with the ratio test.)
If R = 0, the series converges only at z_0. If R = \infty, it converges on all of \mathbb C.