2. Prove that n=1 n -x converges uniformly on S = (√2, ∞o).
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
![x
ex
& fn(x) = x +n for all x 20 (s = [0, 0))
Show for f phoise.
S where f =0
a.
b. Show frf Unif
c. Show for * > f Chrif.
Solna @tix XES =
x
= x+n
Notice fn (x) = x+n, So
(in fri (x)
130
goal
on
=
xtr
#2>0:
X+0
on
[0₁2]
b. Show for f urif. on
>
By AP. I NE N sit / x ¾/2
So 2+N </N LE· Fix X € [0₁2].
L
X 스
x+n
on S.
[0, ∞). Show fr (x) →
2
Then X+n = 2+ (So fn(x) = fm (2))
Assume n>x. Then (fn (x) = f(x)) = /*+₁ -0
2¹
LE
스
2f0
2+1
| fn(x) = f(x) | < E
-
-02E
२६
Cam
Xnº
n->∞ x+n'
f
[0,2]. fix E70
fo is mor
(>N)
if n is large enough
SE
2 & 2² ≤ ¾N CE
스
2+n
2+1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79599c56-a340-49a0-b0ff-829b3947a798%2F1b465e71-3695-4373-bc71-c69cc1dcf2e2%2Fe8xvbb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:x
ex
& fn(x) = x +n for all x 20 (s = [0, 0))
Show for f phoise.
S where f =0
a.
b. Show frf Unif
c. Show for * > f Chrif.
Solna @tix XES =
x
= x+n
Notice fn (x) = x+n, So
(in fri (x)
130
goal
on
=
xtr
#2>0:
X+0
on
[0₁2]
b. Show for f urif. on
>
By AP. I NE N sit / x ¾/2
So 2+N </N LE· Fix X € [0₁2].
L
X 스
x+n
on S.
[0, ∞). Show fr (x) →
2
Then X+n = 2+ (So fn(x) = fm (2))
Assume n>x. Then (fn (x) = f(x)) = /*+₁ -0
2¹
LE
스
2f0
2+1
| fn(x) = f(x) | < E
-
-02E
२६
Cam
Xnº
n->∞ x+n'
f
[0,2]. fix E70
fo is mor
(>N)
if n is large enough
SE
2 & 2² ≤ ¾N CE
스
2+n
2+1
Transcribed Image Text:ove that 2n=1 (n+x)²
2. Prove that in converges uniformly on S = (√2, ∞0).
-x
n=1
3
8
x²
Find the pointwise limit of on S = [1, ∞o). Hint: Defi
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