Uniform Convergence. Optional

15.5 Uniform Convergence. Optional

We know that power series are absolutely convergent (Sec. 15.2, Theorem 1) and, as another basic property, we now show that they are uniformly convergent. Since uniform convergence is of general importance, for instance, in connection with termwise integration of series, we shall discuss it quite thoroughly.

To define uniform convergence, we consider a series whose terms are any complex

functions f 0 (z), f 1 (z), Á

(This includes power series as a special case in which f (z) ⫽ a (z ⫺ z ) m m m 0 .) We assume that the series (1) converges for all z in some region G. We call its sum s (z) and its nth partial sum s n (z); thus

s n (z) ⫽ f 0 (z) ⫹ f 1 (z) ⫹ Á ⫹ f n (z).

Convergence in G means the following. If we pick a z⫽z 1 in G, then, by the definition of convergence at z 1 , for given P⬎0 we can find an N 1 (P) such that

ƒs (z 1 )⫺s n (z 1 )ƒ⬍P

for all n⬎N 1 (P). If we pick a z 2 in G, keeping as before, we can find an P N 2 (P) such that

for all n⬎N 2 (P), and so on. Hence, given an P ⬎ 0, to each z in G there corresponds a number N z (P). This

ƒs (z 2 )⫺s n (z 2 )ƒ⬍P

number tells us how many terms we need (what s n we need) at a z to make ƒs (z) ⫺ s n (z) ƒ smaller than P. Thus this number N z (P) measures the speed of convergence. Small N z (P) means rapid convergence, large N z (P) means slow convergence at the point z considered. Now, if we can find an N (P) larger than all these N z (P) for all z in G, we say that the convergence of the series (1) in G is uniform. Hence this basic concept is defined as follows.

DEFINITION Uniform Convergence

A series (1) with sum s (z) is called uniformly convergent in a region G if for every P⬎0 we can find an N⫽N (P), not depending on z, such that

for all n⬎N (P) and all z in G. Uniformity of convergence is thus a property that always refers to an infinite set in

ƒs (z) ⫺ s n (z) ƒ ⬍ P

the z-plane, that is, a set consisting of infinitely many points.

EXAMPLE 1 Geometric Series

Show that the geometric series 1⫹z⫹z 2 ⫹Á is (a) uniformly convergent in any closed disk ƒ z ƒ ⬉ r ⬍ 1, (b) not uniformly convergent in its whole disk of convergence ƒ z ƒ ⬍ 1.

SEC. 15.5 Uniform Convergence. Optional

Solution. (a) For z in that closed disk we have ƒ1⫺zƒ⭌1⫺r (sketch it). This implies that

1 > ƒ 1 ⫺ z ƒ ⬉ 1>(1 ⫺ r). Hence (remember (8) in Sec. 15.4 with q⫽z )

m⫽n⫹1 a 1⫺r . Since r⬍ 1, we can make the right side as small as we want by choosing n large enough, and since the right

ƒ s(z) ⫺ s n

2 z 2⫽2 1⫺z 2 ⬉

side does not depend on z (in the closed disk considered), this means that the convergence is uniform. (b) For given real K (no matter how large) and n we can always find a z in the disk ƒzƒ⬍1 such that

z n⫹1

ƒzƒ n⫹1

`⫽ ⬎ K 1⫺z , ƒ1⫺zƒ

simply by taking z close enough to 1. Hence no single N (P) will suffice to make ƒs (z) ⫺ s n (z) ƒ smaller than a given P⬎0 throughout the whole disk. By definition, this shows that the convergence of the geometric series in ƒzƒ⬍1 is not uniform.

䊏 This example suggests that for a power series, the uniformity of convergence may at most

be disturbed near the circle of convergence. This is true:

THEOREM 1 Uniform Convergence of Power Series

A power series

with a nonzero radius of convergence R is uniformly convergent in every circular

disk of ƒz⫺z 0 ƒ⬉r radius r⬍R .

PROOF For ƒz⫺z 0 ƒ⬉r and any positive integers n and p we have

⫹Á⫹ ƒa n⫹p ƒr . Now (2) converges absolutely if ƒz⫺z 0 ƒ⫽r⬍R (by Theorem 1 in Sec. 15.2). Hence

⫹Á⫹ a n⫹p (z ⫺ z 0 )

n⫹p

ƒ⬉ƒa n⫹1 ƒr

n⫹1

it follows from the Cauchy convergence principle (Sec. 15.1) that, an P⬎0 being given,

we can find an N (P) such that

1, 2, Á . From this and (3) we obtain ƒa n⫹1 (z ⫺ z 0 ) n⫹1 ⫹Á⫹ a n⫹p (z ⫺ z n⫹p 0 ) ƒ⬍P for all z in the disk ƒz⫺z 0 ƒ ⬉ r, every n⬎N (P), and every p⫽

1, 2, Á . Since N (P) is independent of z, this shows uniform convergence, and the theorem is proved.

䊏 Thus we have established uniform convergence of power series, the basic concern of this

section. We now shift from power series to arbitary series of variable terms and examine uniform convergence in this more general setting. This will give a deeper understanding of uniform convergence.

CHAP. 15 Power Series, Taylor Series