Answers & Hints for Selected Exercises 699- On [O, 2] the limit function is f(x) = 0 if 0 < x::; 2, 1 if x = 0. Then 'in,
llfn -!II = 1, so the convergence is not uniform.
Let 0 < c < 2. Then on [c, 2], llfnll = e-ne:
2
---+ 0 , so fn ---+ 0 uniformly
on [c, 2]. Thus, by Thm. 9.3.8, lim J,2 fn = J,2 f = 0. Then lim f 0
2
n--+oo E E n--+oo J < fn
lim [J; f n + fe:
2
n--+oo f n] ::; n--+oo lim [J;^1 + J,2 E f n] = c. By the forcing principle,
lim f 0
2
n-->oo fn =^0 = J~ f.- On [O, oo), llfnll = 1/n ---+ 0 , so fn ---+ 0 uniformly. But Jt fn = 1/2, so
lim fooo fn =f. fooo f = 0.
n-->oo
n k - Let f(x) = L k(~+l). The radius of convergence is p = 1. Thus, by Cor.
k=l
9.3.12, f is differentiable on (-1, 1), and f'(x) = f; ~<~k.;:) = f; ~::.
k=l k=l - (a) This series is the term-by-term differentiation of the series g(x)
00
L xk = l~x on (-1, 1). Thus, by Cor. 9.3.12, f(x) = g'(x) = lx (1 - x)-^1 =
k=O
(1 - x) -^2 for all x in (-1, 1).
(b) The radius of convergence is 1. Thus, by Cor. 9.3.12, f is differentiable
b ( ) fl ( ) x(^2) x (^3) x (^4) x 5
term-y-term on -1, 1 , so x = T + 2 + 3 + 4 + · · · =
x [f + x 2
2
- ~
3 - x 44 + · · · J = -xln(l - x). Thus, on (-1, 1), J(x) =
- J x ln(l - x)dx. Integrating by parts, f(x) = ~(1 - x^2 ) ln(l - x) + ~x^2 + ~x.
- (a) Each fn is differentiable on IR, and f'(x) = -sin nx.
(b) llfnll = 1/n---+ 0, so fn---+ 0 uniformly on R
(c) 'ix =f. mr (n E Z), {f~(x)} diverges, by Exercise 2.6.20.
(d) This does not contradict Thm. 9.3.11 because one of the hypotheses of
that theorem (uniform convergence of{!~}) is not satisfied. - Let a> 1.
(a) 'ix E [a, oo), Vk E N, kx :::'.'.: ka > 0, so 0 < k~ ::; k^1 a. Thus, Vk E N,
II kl"' II ::; k^1 a and since L k^1 a converges, the Weierstrass M-test guarantees that
L kl"' converges uniformly on [a , oo).
n
(b) Vk E N, let fk(x) = 1/kx = k-x. Consider the series L f~(x)
k=l
n n
- ~ L..,; Ink k X ' Now ' 'ix E [a ' oo) ' 0 < Ink kx -< Ink ka and the series ~ L..,; Ink ka converges
k=l n k=l
by Ex. 8.2.33. Thus, by the Weierstrass M-test, L^1 ~,,k converges uniformly on
k=l n
[a, oo). Therefore, by Thm. 9.3.11, 'ix:::'.'.: a, ('(x) = - L^1 ~,,k.
k=l