1549901369-Elements_of_Real_Analysis__Denlinger_

564 Chapter 9 • Sequences and Series of Functions

Let x =f x 0 in [a,b]. By Equation (4), 3t between x and xo such that

I

[fn(x) - fm(x)] - Un(xo) - fm(xo)] I= lf~(t) _ J:r,(t)I::; llf~ _ J:rill·

x-xo

Thus, from (6), Vx E [a, b] - {xo},

m , n ~ n3 ~ lgn(x) - gm(x)I ::; llf~ -J:nll < €.

Hence, m,n ~ n3 ~ llgn -gmll <con [a,b] - {xo}. Therefore, {gn} converges
uniformly on [a, b] - {xo}, by Theorem 9.2.7.

Part 4: We finish by showing that f is differentiable on [a, b] and Vxo E [a, b],

f'(xo) = lim f~(xo). We first note that from Equation (5), Vx =f xo in [a, b],

n-->oo

lim gn(x) = lim fn(x) - fn(xo) = f(x) - f(xo) = g(x).

n-->oo n-->oo X - XQ X - Xo

Finally, by Theorem 9.3.5, uniform convergence of {gn} on [a, b] - {xo}
allows us to interchange limits, as follows:

f'(xo) = lim g(x), by definition of g;

x-+xo

= lim [ lim gn(x)] , as shown above;

x--+xo n--+oo

= lim [ lim gn(x)] , by Theorem 9.3.5;

n-+oo x-+xo

= lim [!~ ( Xo)], by definition of gn.

n-->oo

Therefore, f is differentiable on [a, b] and Vx E [a, b], f'(x) = lim f~(x). •
n-->oo

Theorem 9.3.11 provides an easy, short proof of the term-by-term differ-
entiability of functions represented by power series. Compare the proof of the
following corollary with that of Theorem 8.6.14.

00
Corollary 9.3.12 Suppose f(x) = 2-:: ak(x - c)k, with radius of convergence
k=O
p > 0. Then Vx E (c - p, c + p), f is differentiable at x and f'(x) =
00
I: akk(x - c)k-l.
k=l
Moreover, the convergence of both series is uniform in [c-R,c+R], where
0 < R < p.

Proof. Exercise 8. •