(a) Let f(x) = e ², x > 0. Show that, for every n ≥ 1, the n'th derivative f(n)(x) is of the form Pn(1/x) e for some polynomial P₁ (depending on n). · (b) Define 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 g of part (b) is not equal to the sum of its Maclaurin series. h→0+ (c)
(a) Let f(x) = e ², x > 0. Show that, for every n ≥ 1, the n'th derivative f(n)(x) is of the form Pn(1/x) e for some polynomial P₁ (depending on n). · (b) Define 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 g of part (b) is not equal to the sum of its Maclaurin series. h→0+ (c)
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¯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→∞](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78284481-d415-4f9a-b48b-4bb6d9c066f4%2F4375d8c1-cda1-48b2-abb9-df062af984c2%2Fpikkhy_processed.png&w=3840&q=75)
Transcribed Image Text:(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→∞
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