13.3 Power Series
Remark 13.2.2 Every uniformly convergent sequence is pointwise conver-
gent. If (/„) converges pointwise on E, then there exist a function f such
that, for every e > 0 and for every x £ E, there is an integer N, depending
on e and x, such that |/„(x) — f{x)\ < e holds if n > N; if (/„) converges
uniformly on E, it is possible, for each e > 0, to find one integer N which will
do for all x £ E.
Proposition 13.2.3 (Cauchy's uniform convergence) A sequence of func-
tions, (fn), defined on E, converges uniformly on E if and only if
Ve >0, 3N eN3m>N,n>N,x e E=> \fm{x) - fn{x)\ < e.
Corollary 13.2.4 Suppose linin-yoo fn{x) = /(x), x G E. Put
M„ = sup|/n(x)-/(x)|.
xeE
Then, fn —> / uniformly on E if and only if Mn —>• 0 as n —>• oo.
Proposition 13.2.5 (Weierstrass) Suppose (/„) is a sequence of functions
defined on E, and |/(x)| < M„, x € E, n = 1, 2, 3,... Then, ^ fn converges
uniformly on E if ^2 Mn converges.
Proposition 13.2.6
lim lim fn{t) = lim lim fn(t).
t~¥x n—+oo n—>oo t—>x
Remark 13.2.7 The above assertion means the following: Suppose /„—>•/
uniformly on a set E in a metric space. Let x be a limit point of E, and
suppose that limt>x/n(i) —> An, n = 1,2,3... Then, (An) converges, and
lim^a f(t) = limn>oo An.
Corollary 13.2.8 // (/„) is a sequence of continuous functions on E, and if
fn —> f uniformly on E, then f is continuous on E.
Remark 13.2.9 The converse is not true. A sequence of continuous func-
tions may converge to a continuous function, although the convergence is not
uniform.
13.3 Power Series
Definition 13.3.1 The functions of the form
oo
f(
x
) = YlCnxU
n=0
or more generally,
oo
f(
x
) = Y2cn(x-a)r
ra=0
are called analytic functions.