9.3 Implications of Uniform Convergence in Calculus 567- Evaluate each of the following integrals and justify your answers:
( ) r ~ cos kx d (b) r ~ sink2kx dx ( ) (
2
~ kx
a lo ~ ~ x lo ~ c 11 ~ ke- dxn n^2 - -- Use the functions fn(x) = over the interval
{
l. - -"'--if 0 < x < n }
0 if x > n
(0, +oo) to show that Theorem 9.3.8 does not apply over infinite intervals.
That is, uniform convergence of {Jn} does not always preserve (improper)
integrability over infinite intervals.- Prove Corollary 9.3.12.
- Use Corollary 9.3.12 to show that V lxl < 1 .!!:_ ( f xk )
'dx k= 1 k(k+l)
00 k-1
2:-x-.
k=l k + 1
. 00 (-l)k - Use Corollary 9.3.12 to prove that the function y(x) = k~O ~x^3 k is a
solution of the differential equation y' + 3x^2 y = 0 on ( -oo, oo).
- Use results of this section to find the sums of the following series, for
xE(-1,1):
(a) 1+2x + 3x^2 + 4x^3 + · · ·. [Think geometric series.]
x3 x4 x5 x6
(b) t-:3 +
2
.
4+
3.
5+
4.
6+ · · ·. [See Equation (22) of Example 8.6.18.]- Show that { .j x^2 + ~} converges uniformly to lxl on R Verify that each
function in the sequence is differentiable everywhere, but the limit func-
tion is not.
cosnx.- For all n EN, define fn(x) = --on (-00,00). Prove that Un} is
n
a sequence of differentiable functions on (-oo, oo) converging uniformly
to 0 on ( - oo, oo), but {f~ ( x)} diverges except when x is an integral
multiple of 1r. [See Exercise 2.6.20.] Doesn't this contradict Theorem
9.3.11? Explain. - Explain what happens if you try to redo Example 9.3. 13 using the fol-
lowing functions instead of the one given there?
00 00
(b) L sin I (c) 2: cos I
k=l k=l