1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

7.1 • UNIFORM CONVERGENCE 251

y

10

8 f<Xo)

6

/

,, ,/

=

======~~-~~~~...- ,,

-------

-'=--=--=-"'-"'-"'-"'-=-=-=-=-::".!-!:-:-"'-"'-=----=F----.._----.--~ x

• I --0.5 \ 0.5
  y=j(x)-£
  Figure 7.1 The geometric series does not converge unilormly on (-1, 1).

such that, no matter how large n is, we can find xo E ( -1, 1) with the property
that Sn (xo) lies outside this bandwidth. In other words, Figure 7.1 illustrates
the negation of Statement (7-2), which in technical terms we state as:

There exists e > 0 such that, for all positive integers N,

there are some n ;::: N and some zo E T

such that ISn (ZQ) -f (Z<>)I 2: e.

(7-3)

In the exercises, we ask you to use Statement (7-3) to show that the partial

sums of the geometric series do not converge uniformly to f (z) = 1 ~ .. for points

z E D1 (O).

A useful procedure known as t he Weierstrass M-test can help determine

whether an infinite series is uniformly convergent.