556 Chapter 9 • Sequences and Series of Functions
- Determine whether the following sequences {1 n} converge uniformly on
[O, +oo):
(a) fn(x) = xne-nx
1
(c) ln(x) = nx2 + 1
(b) ln(x) = ~e-x/n
n 1
(d) ln(x) = n(x2 + 1)- For each of the sequences Un} in Exercise 9.1.3, determine whether Un}
converges uniformly on its set S of pointwise convergence. If it doesn't,
find a subset of S on which {1 n} does converge uniformly, if possible. - Prove Part 1 of Theorem 9.2.7.
- Prove that if Un} is a sequence in B(S) converging uniformly on S to 1,
then lllnll __.., 11111· [First prove that I lllnll - 111111 :S llln -111.] - Suppose Un}, {gn} converge uniformly on S , and let r ER Prove that
(a) Un+ 9n} converges uniformly on S.
(b) { r 1 n} converges uniformly on S.
(c) if Un} and fon} are in B(S), then {ln9n} converges uniformly on S. - Prove Theorem 9.2.11.
- Prove Corollary 9.2. 12
15. Prove Corollary 9.2.15.16. Consider the functions lk(x) =. Prove that on [O, 1]
{1 if k~l < x ::::: -k }
0 otherwise
00
the series L fk converges pointwise (absolutely) but not uniformly.
k=O- Prove that the given series converges uniformly but not absolutely on the
given interval.
'I,
oo ( l)k+lxk
(a) L - k on [O, 1]
00 (-l)k
(b) L ~ on (-00,00)
k=l k=l x +.- Modify the proof of Theorem 8.5.3 to obtain a proof of Theorem 9.2.16.
- Determine whether the following series converge uniformly on the indi-
cated set:
( )
~ sinkx 1Tl>
a L,, k 2 on JI">.
k=loo k x k
(b) I: -k - on (0, 1)
k=l + 100
(c) L e-kx on (0, oo)
k=O
00 00 00 1
(d) L e-kx on [1, oo) (e) L xke-kx on [O, oo) (f) L k 2 2 on (0 , 1]
k=O k=l k=O 1 + X