Special Assignment #4 โ Solutions Math 315 March 9, 2005
1. Consider the sequence (fn) defined on [0,+โ) by
fn(x) = nx
1 + nx.
(a) Find the pointwise limit fof (fn) on [0,+โ).
f(x) = lim
nโโ
fn(x) = ๎0,if x= 0;
1,if x > 1; for all xโ[0,โ).
(b) Show that this convergence is not uniform.
Since each fnis continuous on [0,โ), but fis not, the convergence is not unifrom.
(c) However, prove that the sequence (fn) converges uniformly on [a, +โ) for every a > 0.
Let a > 0 and let ฮต > 0. Set N= 1/aฮต. Then for all xโ[a, โ),
n>N=โ |fn(x)โ1|=1
1 + nx <1
nx โคโ1
na <1
Na =ฮต.
2. Consider the series โ
X
n=1
sin nx
3nxn.
Show that the series converges uniformly on [โa,a] for every 0 < a < 3. [Hint: 25.7.]
The series Pan/3nconverges since it is geometric with a/3<1. But also,
๎
๎
๎
๎
sin nx
3n๎
๎
๎
๎
โค|x|n
3nโคan
3n
for all nโlN and all xโ[โa, a]. So by the M-test, the given series converges
uniformly on [โa, a].