(a) Let f(x) = e¯¹², x x > 0. Show that, for every n ≥ 1, the n'th derivative f(n)(x) is of the form P₁(1/x). e2 for some polynomial P₁ (depending on n). (b) Define (c) g(x) = 0 e - if x ≤ 0 if x > 0. Use part (a) to prove that g(n) (0) = 0 for all n ≥ 1. [Hint: You may want to use the fact that lim F(1/h) : lim F(t), for any function F.] Conclude that function 9 of part (b) is not equal to the sum of its Maclaurin series. h→0+ t→∞
(a) Let f(x) = e¯¹², x x > 0. Show that, for every n ≥ 1, the n'th derivative f(n)(x) is of the form P₁(1/x). e2 for some polynomial P₁ (depending on n). (b) Define (c) g(x) = 0 e - if x ≤ 0 if x > 0. Use part (a) to prove that g(n) (0) = 0 for all n ≥ 1. [Hint: You may want to use the fact that lim F(1/h) : lim F(t), for any function F.] Conclude that function 9 of part (b) is not equal to the sum of its Maclaurin series. h→0+ t→∞
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
![(a)
Let f(x) = e¯2²2²2, x > 0. Show that, for every n ≥ 1, the n'th derivative f(n) (x) is of
the form P₁(1/x) e for some polynomial P₁ (depending on n).
·
(b)
Define
(c)
g(x):
=
0
12
if x < 0
if x > 0.
Use part (a) to prove that g(n) (0) = 0 for all n ≥ 1.
h→0+
t→∞
[Hint: You may want to use the fact that lim F(1/h) = lim F(t), for any function F.]
Conclude that function g of part (b) is not equal to the sum of its Maclaurin series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbea85843-c685-4611-8e57-d8b3fe40f7ea%2F4ef5ae86-1bf1-4c23-9980-9f70fcc1c3cf%2Fpxjnap9_processed.png&w=3840&q=75)
Transcribed Image Text:(a)
Let f(x) = e¯2²2²2, x > 0. Show that, for every n ≥ 1, the n'th derivative f(n) (x) is of
the form P₁(1/x) e for some polynomial P₁ (depending on n).
·
(b)
Define
(c)
g(x):
=
0
12
if x < 0
if x > 0.
Use part (a) to prove that g(n) (0) = 0 for all n ≥ 1.
h→0+
t→∞
[Hint: You may want to use the fact that lim F(1/h) = lim F(t), for any function F.]
Conclude that function g of part (b) is not equal to the sum of its Maclaurin series.
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why here f(x)=e^(x^2), support to be f(x)=e^(-1/(x^2))
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