## Analysis I: Lecture 15

In which we study the radius of convergence of a complex power series.

• Theorem 31 (Existence of a radius of convergence) Let $\sum\limits_{n=0}^{\infty} a_n z^n$ be a complex power series.  Then either the series converges absolutely for all $z \in \mathbb{C}$, or there is a non-negative real number $R$ such that the series converges absolutely if $|z| < R$ and diverges if $|z| > R$.  To prove this, we ‘guessed’ a suitable value of $R$, namely the supremum of $\{x \in \mathbb{R} : x \geq 0 \textrm{ and } \sum\limits_{n=0}^{\infty} a_n x^n \textrm{ converges}\}$ (when this makes sense).  We repeatedly used Lemma 30 to show that this value of $R$ does the job.
• Definition of the radius of convergence and circle of convergence of a complex power series.
• Lemma 32 Let $\sum\limits_{n=0}^{\infty} a_n z^n$ have radius of convergence $R$.  Then $\sum\limits_{n=1}^{\infty} n a_n z^{n-1}$ also has radius of convergence $R$.
• I gave out the third examples sheet.

#### Understanding today’s lecture

Investigate some more examples of power series.  Given a power series, can you find its radius of convergence?  Turning that around (a very good idea when trying to understand what’s going on), given a non-negative real number $R$, can you find a power series with radius of convergence $R$?  Can you find a power series with radius of convergence $R$ that converges everywhere on its circle of convergence?  That diverges everywhere on its convergence?  That converges at some but not all points on its circle of convergence?

Can you finish the example from the notes: at which points does $\sum\limits_{n=1}^{\infty} \frac{z^n}{n}$ converge?