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?

Further reading

This is a classic topic for introductory analysis and calculus books.  What are your favourite examples?  There’s an interesting historical perspective in A Radical Approach to Real Analysis by David Bressoud: this is a textbook, but it presents the material as it was studied originally.  So if you’re interested in learning something about why the subject has developed as it has, then this would be a great place to look.

The examples sheet includes the names of a couple of mathematicians that I haven’t mentioned in lectures, so here are handy links to their biographies: Darboux and l’Hôpital (worth reading just to see the length of his full name!).

Preparation for Lecture 16

Can you show that a power series is differentiable everywhere within its radius of convergence, and that its derivative is the power series that it really ought to be?

How do we define the exponential function?  What properties do we expect it to have?  Can you deduce those properties from the definition you have chosen?

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