Series of Complex Functions

Series of Functions

A particular important class of sequence of functions are series of functions. The series refers to the limit of sequence of partial sums , and the series is said to converge pointwise/uniformly/normally if the sequence of partial sums converges pointwise/uniformly/normally.

Absolute Convergence of Series of Functions

A series of functions is said to converge absolutely on a set if it converges pointwise absolutely, that is, if the series converges for each .

Theorem

We have the following properties for complex series:

1. Let be an absolutely convergent series. Then any reodering of the series converges to the same value.
2. If converges, then
3. If both and converge absolutely, then

Weierstrass M-test for Numbers

Let be a sequence, and suppose are nonnegative numbers such that for all and such that converges, then converges.

Proof We’re aiming to show that partial sum is Cauchy. Let , and let . Pick such that implies that . Then for we have Hence the sequence of partial sums is Cauchy and therefore convergent.

Weierstrass M-test for Functions

Let be a set, and let be a sequence of functions indexed by . Let be a sequence of nonnegative real numbers such that for all and converges. Then the series converges uniformly and absolutely on .

Proof Apply Weierstrass M-test for numbers to the series to see that converges absolutely. Now define . We just need to check uniformly. Let , , . Since converges, we can find such that implies that . Then for we have Hence now take the limit as to see that Clearly did not depend on , so the convergence is uniform.

Remark

Note that the M-test can only be used on domains where you expect to get uniformly convergence. If you expect to get normal convergence but not uniform convergence, you have to apply the M-test on subdomains. The following example is typical: e.g. Let be a sequence with . Then converges normally and absolutely on . Clearly in the case that the corresponding series does not converge uniformly on . But we can consider the subdomain for and apply the M-test to show that the series converges uniformly on . Since every lies in some , this shows that the series converges normally on .

Power Series

Power Series

A series of the form is called a power series centered at .

Abel's Theorem

If the power series converges pointwisely for fixed , then it converges absolutely and normally on .

Proof Fix some arbitrary and consider the disk . Since converges, we have so for some . Then for any , . Hence by M-test, converges uniformly and absolutely on , thus it converges normally and absolutely on .

Corollary

For any power series there is a number such that the series converges absolutely and normally on and diverges when .

Proof Let . Let . By definition of supremum, we may choose a sequence such that . By Abel’s theorem, the series converges absolutely and normally on for all , so it converges absolutely and normally on . On the other hand, if , then , so the series diverges.

Remark

Note that and . This number is called the radius of convergence of the power series, and the disk is called the disk of convergence.

Proposition

For any , there is a power series with radius of convergence .

Proof The power series has radius of convergence , as it diverges. Any polynomial has radius of convergence , as polynomial is finite, therefore converges anywhere. For , the power series has the radius of convergence .

Theorem

Let be a power series, then its radius of convergence is

Proof Notice that By the root test, the series converges if and diverges if . That is if the series converges and if the series diverges, yielding the desired result.