3. Let fn [0, π] → R be defined by : (a) fn(2)= sin na nx n' 3 if x # 0 otherwise. Show that fn(x) → 0 as n→∞o for all x = [0, π]. : (b) Find the uniform norm ||fn|| (0,7] of fn on [0, π]. (c) Show that fn converges uniformly to 0 on the subinterval [8, π] as n→ ∞o for any 0 < d < T, but does not converge uniformly to 0 on [0, π].
3. Let fn [0, π] → R be defined by : (a) fn(2)= sin na nx n' 3 if x # 0 otherwise. Show that fn(x) → 0 as n→∞o for all x = [0, π]. : (b) Find the uniform norm ||fn|| (0,7] of fn on [0, π]. (c) Show that fn converges uniformly to 0 on the subinterval [8, π] as n→ ∞o for any 0 < d < T, but does not converge uniformly to 0 on [0, π].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![3. Let fn [0, π] → R be defined by
$n (2) =
(a)
(b)
sin na
3
na
n'
Show that fn(x) →0 as n →∞o for all x = [0, π].
Find the uniform norm ||fn|| (0,r) of fn on [0, π].
(c)
Show that fn converges uniformly to 0 on the subinterval
[S, π] as n→ ∞o for any 0 << T, but does not converge uniformly to
0 on [0, π].
(d)
lim
8-8
if x #0
otherwise.
Using part (c) and your result in part (b), show that
sin nx
nx
-dx = 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1652026a-6ba6-42d7-9e35-ad3e7516e1ec%2F18454b05-ccf3-460a-b608-0e5038881b2d%2F6ucdg0f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Let fn [0, π] → R be defined by
$n (2) =
(a)
(b)
sin na
3
na
n'
Show that fn(x) →0 as n →∞o for all x = [0, π].
Find the uniform norm ||fn|| (0,r) of fn on [0, π].
(c)
Show that fn converges uniformly to 0 on the subinterval
[S, π] as n→ ∞o for any 0 << T, but does not converge uniformly to
0 on [0, π].
(d)
lim
8-8
if x #0
otherwise.
Using part (c) and your result in part (b), show that
sin nx
nx
-dx = 0.
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