562 Chapter 9 • Sequences and Series of FunctionsSolution: Let 0 < a < n. On the interval [a, n], II si~;x II < ~a -t 0.
sinnx.
Thus, ---t 0 umformly on [a, n]. So, by Theorem 9.3.8,
nx
lim 1,,. sinnx dx = 1,,. ( lim sinnx) dx = 1,,. 0 = 0. D
n-+oo a nx a n-+oo nx aUniform convergence does not necessarily preserve (improper) integrals
over infinite intervals. See Exercise 7 for an example.
The relationship between uniform convergence and derivatives is not so
straightforward. As we saw in Example 9.3.4(a), uniform convergence does not
preserve differentiability. Even when a sequence Un} of differentiable functions
converges uniformly to a differentiable limit function f, the sequence U~(xo)}
might not converge to f'(x 0 ). (See Exercise 13.) The following theorem shows
that for sequences of differentiable functions, the focus shifts from uniform
convergence of Un} to that of{!~}.
Theorem 9.3.11 (Uniform Convergence and Differentiation) Suppose
{f n} is a sequence of differentiable functions on [a, b] such that {f~} converges
uniformly on [a, b], and suppose that {fn(xo)} converges for at least one xo E
[a, b]. Then {f,,J converges uniformly on [a, b] to a differentiable function f,
and Vx E [a, b], f'(x) = lim f~(x).
n-+oo
Proof. Suppose {fn} and U~} satisfy the hypotheses. Then a< b.
Part 1: We first show that {fn} converges uniformly on [a, b]. Let c: > 0.
Since {!~} converges uniformly on [a, b], the uniform Cauchy criterion tells
us that :ln1 EN 3
m, n?. n1 '* Iii~ -J:nll < 2 (b ~a)"
Let xo be any point of [a, b] for which Un(xo)} converges. By the Cauchy
criterion for sequences of real numbers, :J n 2 E N 3
Suppose m,n?. max{n1,n2}. Consider any x =f=. xo in [a,b]. By the mean
value theorem applied to the function f n - f m on the closed interval between
x and xo, :J t between x and xo such that
[fn(x) - fm(x)] - [fn(xo) - fm(xo)] = (x - xo) [f~(t) - J:n(t)], (4)
fn(x) - fm(x) = fn(xo) - fm(xo) + (x - xo) [f~(t) - J:n(t)].