210 Chapter4
for every n, i.e., the two power series have the same coefficients and, as a
consequence, R 1 = R2.
Proof Since f(z) and g(z) are continuous functions on JzJ :::; o, from
f(zk) = g(zk) we get
lim f(zk) = lim g(zk)
k->oo k->oo
or f(O) = g(O)
i.e., a 0 = b 0 • Hence we have
aizk + a2z~ + · · · = b1zk + b2z~ + · · ·
and since Zk =f-0, it follows thatai + a2Zk + · · · = b1 + b2zk + · · · (4.11-11)
Each of the series in ( 4.11-11) has the same radius of convergence as
the corresponding original series, so, with z in place of Zk, they define
continuous functions fi(z) and g 1 (z) on JzJ :::; o. From (4.11-11) we have
fi(zk) = gi(zk) and letting k ---> oo, we get
orAfter repeating the process m times we obtain am = bm, and if we let
fm+1(zk) = am+l + am+2Zk + · · ·
gm+1(zk) = bm+l + bm+2Zk + · · ·
from fm+1(zk) = gm+1(zk) we conclude, as before, that fm+1(0) =
gm+1(0), or am+l = bm+l· Thus, by mathematical induction, we have
an = bn for every n-.
Exercises 4.2
- Test each series for convergence.
(a) ~ [ ~2 + (n + lXn + 2) i]
(c) ~ [n-^3 + (-l)n ~i]
- Test for absolute convergence.
00
(a) 2)n-^1 + n-^2 i)
1
00
(c) L:(e-n - rni)
1 - Test for absolute convergence.
(b) ~ (n-
1
1
2
_ ~! i)(d) ~[n\n + n(n~l)i]
00