2. Let fn: R→ R where fn(x) = ²+nz for all n E N and let f: R → R where f(x) = x. n (a) Graph f1, f2, f3, and f on the same axes. (b) Prove that fn converges pointwise to f on R. (c) Does fn converge uniformly to f on R? Prove your answer. (d) Does fn converge uniformly to f on [-1, 1]? Prove your answer.
2. Let fn: R→ R where fn(x) = ²+nz for all n E N and let f: R → R where f(x) = x. n (a) Graph f1, f2, f3, and f on the same axes. (b) Prove that fn converges pointwise to f on R. (c) Does fn converge uniformly to f on R? Prove your answer. (d) Does fn converge uniformly to f on [-1, 1]? Prove your answer.
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
![2. Let fn: R→ R where fn(x) = ²+n for all n E N and let f: R → R
where f(x) = x.
n
(a) Graph f1, f2, f3, and f on the same axes.
(b) Prove that fn converges pointwise to f on R.
(c) Does fn converge uniformly to f on R? Prove your answer.
(d) Does fn converge uniformly to f on [-1, 1]? Prove your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79599c56-a340-49a0-b0ff-829b3947a798%2Fea7082f3-b3a2-4280-8d6f-7132a22b82cd%2Fi2kknr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Let fn: R→ R where fn(x) = ²+n for all n E N and let f: R → R
where f(x) = x.
n
(a) Graph f1, f2, f3, and f on the same axes.
(b) Prove that fn converges pointwise to f on R.
(c) Does fn converge uniformly to f on R? Prove your answer.
(d) Does fn converge uniformly to f on [-1, 1]? Prove your answer.
![ex fn(x) = x+n for all x ₂₂0 (S = [0, 0))
a. Show for ⇒ f phoise
S where of 20
b. Show fri f Unif
c. Show for * f Chrif.
*
= x+n
Sula - @ Fix xes = [0,00). Show fr (x) -
Nokice fn (x) = x+n, So
Cem for (x) = Cam
130
7-10
gord | fr. (x) = f(x) | <.
goal
-
√x+n-0
२६
+2>0:
6 Show for f urif. on
[0₁2]. fix E₂0
b.
By
2
AP. I NE α s-t¼/μ< ¾//2
So R+N <³ <E· Fix x € [0₁2].
N
Then X+1 = 2+ (So fn(x) ≤ fn (2))
Assume no N. Then (fr (x) = f(x)) = √x+n
x+7-0
2
2
<
LE
2+0
2+M
Me
_X
on
x+n
on
[0₁2]
X+0
S
on S.
хоо
x+n'
fo is micr
(^>if)
if n is large enough.
<E
2 = 2AN = ¾/N CE
스
2+0
Ĉ](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79599c56-a340-49a0-b0ff-829b3947a798%2Fea7082f3-b3a2-4280-8d6f-7132a22b82cd%2Fnu1smho_processed.jpeg&w=3840&q=75)
Transcribed Image Text:ex fn(x) = x+n for all x ₂₂0 (S = [0, 0))
a. Show for ⇒ f phoise
S where of 20
b. Show fri f Unif
c. Show for * f Chrif.
*
= x+n
Sula - @ Fix xes = [0,00). Show fr (x) -
Nokice fn (x) = x+n, So
Cem for (x) = Cam
130
7-10
gord | fr. (x) = f(x) | <.
goal
-
√x+n-0
२६
+2>0:
6 Show for f urif. on
[0₁2]. fix E₂0
b.
By
2
AP. I NE α s-t¼/μ< ¾//2
So R+N <³ <E· Fix x € [0₁2].
N
Then X+1 = 2+ (So fn(x) ≤ fn (2))
Assume no N. Then (fr (x) = f(x)) = √x+n
x+7-0
2
2
<
LE
2+0
2+M
Me
_X
on
x+n
on
[0₁2]
X+0
S
on S.
хоо
x+n'
fo is micr
(^>if)
if n is large enough.
<E
2 = 2AN = ¾/N CE
스
2+0
Ĉ
Expert Solution
Step 1
(a) The graphs of f_1,f_2,f_3,f_4 is plotted below.
Step by step
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