7.1 • UNIFORM CONVERGENCE 255which becomes
00 00- Lo g ( 1 - zo ) = ~ L., --z^1 0 n+l = ~ L., - z^1 0 n.
n=O n+l n = l n
The point Zo E Di (0) was arbitrary, so we are done.
-------~EXERCISES FOR SECTION 7.1
- This exercise relates to Figure 7 .1.
(a) For x near - 1, is the graph of S., (x) above or below f (x)? Explain.
(b) Is the index n in Sn (x) odd or even? Explain.
(c) Assuming that the graph is accurate to scale, what is the value of n in
Sn (x)? Explain.- Complete the details to verify t he claim of Example. 7.1.
- Prove that the following series converge uniformly on the sets indicated.
00
(a) E -bzk on D 1 (0) = {z: lzl $ 1 }.
k=l
00
(b) k~O (»~i)" on {z: lzl::::: 2}.
00 • -
(c) L; ,•1+ 1 on D, (0), where 0 < r < 1.
k = O
n - 14. Show that Sn (z) = L; zk =^1 ;_:; does not converge uniformly to f (z) = 1 ~.
k =O
on the set T = D1 (0) by appealing to Statement (7-3). Hint: Given e > 0 and a
positive integer n, let Zn = e ~.- Why can't we use the arguments of T heorem 7. 2 to prove that the geometric series
converges uniformly on all of D 1 (O)? - By starting with the series for t he complex cosine given in Section 5.4, choose an
appropriate contour and use the method in Example 7 .2 to obtain the series for
the complex sine. - Suppose that the sequences of functions {f.,} and {g,.} converge uniformly on the
set T.
(a) Show t hat the sequence {f., + g.,} converges uniformly on the set T.
(b) Show by example that it is not necessarily the case that {!,. g,,} con-
verges unjformly on the set T. - On what por tion of D1 (0) does the sequence {nz"}::°=t converge, and on what
portion does it converge uniformly?