Uniform Convergence of Series of Functions

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Recall that a sequence of functions with common domain is said to be pointwise convergent if for all and for all there exists an such that if then:

Also recall that a sequence of functions with common domain is said to be uniformly convergent if for all there exists an such that if then for all we have that:

We will now extend the concept of pointwise convergence and uniform convergence to series of functions.

Definition: Let be a sequence of functions with common domain . The corresponding series is said to be Pointwise Convergent to the sum function if the corresponding sequence of partial sums (where ) is pointwise convergent to .

For example, consider the following sequence of functions defined on the interval :

We now that this series converges pointwise for all since the result series is simply a geometric series to the sum function .

Definition: Let be a sequence of functions with common domain . The corresponding series is said to be Uniformly Convergent to the sum function if the corresponding sequence of partial sums is uniformly convergent to .

The geometric series given above actually converges uniformly on , though, showing this with the current definition of uniform convergence of series of functions is laborious. We will soon develop methods to determine whether a series of functions converges uniformly or not without having to brute-force apply the definition for uniform convergence for the sequence of partial sums.

Cauchy's Uniform Convergence Criterion for Series of Functions

If we have a sequence of functions with common domain then the corresponding series of functions is said to be uniformly convergent if the corresponding sequence of partial sums is a uniformly convergent sequence of functions.

We will now look at a nice theorem known as Cauchy's uniform convergence criterion for series of functions.

Theorem 1: Let be a sequence of real-valued functions with common domain . Then is uniformly convergent on if and only if for all there exists an such that if and for all we have that for all .

  • Proof: Suppose that is uniformly convergent to some limit function on . Let denote the sequence of partial sums for this series. Then we must have that uniformly on . So, for there exists an such that if and for all we have that:

  • For any let . Then and so:

  • Suppose that for all there exists an such that if and for all we have that:

  • Let . Assume without loss of generality that and that for some . Then from above we see that for all :

  • So converges uniformly by the Cauchy uniform convergence criterion for sequences of functions. So converges uniformly on .

The Weierstrass M-Test for Uniform Convergence of Series of Functions

Recall that if is a sequence of real-valued functions with common domain , then we say that the corresponding series of functions is uniformly convergent if the sequence of partial sums is a uniformly convergent sequence.

We will now look at a very nice and relatively simply test to determine uniform convergence of a series of real-valued functions called the Weierstrass M-test.

Theorem 1: Let be a sequence of real-valued functions with common domain , and let be a sequence of nonnegative real numbers such that for each and for all . If converges then uniformly converges on .

  • Proof: Suppose that there exists a sequence of nonnegative real numbers such that for all and for all we have that:

  • Furthermore, suppose that converges to some , . Then we have that for all :

  • So the converges for each by the comparison test.

Weierstrass Example

The series expansion of the exponential function can be shown to be uniformly convergent on any bounded subset using the Weierstrass M-test.

Theorem (Weierstrass M-test). Let be a sequence of functions and let be a sequence of positive real numbers such that for all and If converges, then converges uniformly on .

The complex exponential function can be expressed as the series:

Any bounded subset is a subset of some disc of radius centered on the origin in the complex plane. The Weierstrass M-test requires us to find an upper bound on the terms of the series, with independent of the position in the disc:

To do this, we notice

and take

If is convergent, then the M-test asserts that the original series is uniformly convergent.

The ratio test can be used here:

which means the series over is convergent. Thus the original series converges uniformly for all and since , the series is also uniformly convergent on

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