5. Let P([0, 1]) be the space of polynomials on [0, 1] with the norm of uniform convergence, that is, for p E P, ||p|| max p(x). €[0,1] Let pn(x) = P([0, 1]) with n E N be the following sequence of polynomials xn = x² Pn(x) = = 1+x+ + + 2! n! Show pn(x) is a Cauchy sequence. Show this and determine whether it converges to an element of P([0, 1]).
5. Let P([0, 1]) be the space of polynomials on [0, 1] with the norm of uniform convergence, that is, for p E P, ||p|| max p(x). €[0,1] Let pn(x) = P([0, 1]) with n E N be the following sequence of polynomials xn = x² Pn(x) = = 1+x+ + + 2! n! Show pn(x) is a Cauchy sequence. Show this and determine whether it converges to an element of P([0, 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
![5. Let P([0, 1]) be the space of polynomials on [0, 1] with the norm of uniform convergence,
that is, for p EP,
||p|| = max |p(x).
x[0,1]
Let pn(x) = P([0, 1]) with n € N be the following sequence of polynomials
xn
n!
x²
Pn(x) = 1 + x + +
2!
Show pn(x) is a Cauchy sequence. Show this and determine whether it converges to an element
of P([0, 1]).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb502ef85-41e9-403a-865d-ccc6eb209782%2F64a4c63c-3803-4498-9f6b-0d5f6f6601a3%2F3um8fnt_processed.png&w=3840&q=75)
Transcribed Image Text:5. Let P([0, 1]) be the space of polynomials on [0, 1] with the norm of uniform convergence,
that is, for p EP,
||p|| = max |p(x).
x[0,1]
Let pn(x) = P([0, 1]) with n € N be the following sequence of polynomials
xn
n!
x²
Pn(x) = 1 + x + +
2!
Show pn(x) is a Cauchy sequence. Show this and determine whether it converges to an element
of P([0, 1]).
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