Question 2. For n ≥ 1 and x € R, let fn(2) = xn n + x²n and f(x) = f(x). Let r [0, 1) and sn (r) = sup n=1 n wp{|ƒ(x) - Σ fn(x)| : −1≤ x ≤r}. k=1 (a) Prove that sn(r) →0 as n→∞. (b) Prove that Σ fn(x) → f(x) uniformly on [-1, r]. (c) Prove that fn(x) does not converge uniformly to f(x) on (-1,1).
Question 2. For n ≥ 1 and x € R, let fn(2) = xn n + x²n and f(x) = f(x). Let r [0, 1) and sn (r) = sup n=1 n wp{|ƒ(x) - Σ fn(x)| : −1≤ x ≤r}. k=1 (a) Prove that sn(r) →0 as n→∞. (b) Prove that Σ fn(x) → f(x) uniformly on [-1, r]. (c) Prove that fn(x) does not converge uniformly to f(x) on (-1,1).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![Question 2. For n ≥ 1 and x = R, let
fn(2)
=
Let r = [0, 1) and sn (r)
xn
n + x²n
and
= sup
Sn
∞
f(x) = Σ fn(x).
n=1
n
p{\f(x) – Ë fn(x)| : −1 ≤ x ≤r}.
k=1
(a) Prove that sn (r) → 0 as n →∞.
(b) Prove that Σ fn(x) → ƒ(x) uniformly on [−1, r].
(c) Prove that fn(x) does not converge uniformly to f(x) on (−1,1).
(d) Is f continuous at x = −1?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffef45238-eb5b-4e87-99b3-bef83613c2a6%2F7de7b55c-d42c-4848-b2b2-b6e1f9c76590%2Fld5gl34_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 2. For n ≥ 1 and x = R, let
fn(2)
=
Let r = [0, 1) and sn (r)
xn
n + x²n
and
= sup
Sn
∞
f(x) = Σ fn(x).
n=1
n
p{\f(x) – Ë fn(x)| : −1 ≤ x ≤r}.
k=1
(a) Prove that sn (r) → 0 as n →∞.
(b) Prove that Σ fn(x) → ƒ(x) uniformly on [−1, r].
(c) Prove that fn(x) does not converge uniformly to f(x) on (−1,1).
(d) Is f continuous at x = −1?
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