Show detailed solution for all. a) Provide the Maclaurin series expansion of f(x) In(3 – 2x) b) Find its interval of convergence Example way of solving: The function f (r) 2" has a Maclaurin series expansion. Solving for f(n) (0), f(0) (0) = f(0) (x) = 2" f(1) (x) = In 2 2" f(2) (a) = (In 2)2 . 2" f(1) (0) = In 2 f(3) (x) = (In 2) - 2" f(4) (x) = (In 2)* - 2" f(2) (0) = (In 2)2 f(3) (0) = (In 2)* f(4) (0) = (In 2)* For n 2 0, f(m) (x) = (In 2)" - 2" f(n) (0) = (In 2)" So, the Maclaurin series expansion of f (a) = 2" is f(n) (0) f (r) %3D n! n=0 (In 2)" n! n=0 The interval of convergence of this power series is (-, +x).
Show detailed solution for all. a) Provide the Maclaurin series expansion of f(x) In(3 – 2x) b) Find its interval of convergence Example way of solving: The function f (r) 2" has a Maclaurin series expansion. Solving for f(n) (0), f(0) (0) = f(0) (x) = 2" f(1) (x) = In 2 2" f(2) (a) = (In 2)2 . 2" f(1) (0) = In 2 f(3) (x) = (In 2) - 2" f(4) (x) = (In 2)* - 2" f(2) (0) = (In 2)2 f(3) (0) = (In 2)* f(4) (0) = (In 2)* For n 2 0, f(m) (x) = (In 2)" - 2" f(n) (0) = (In 2)" So, the Maclaurin series expansion of f (a) = 2" is f(n) (0) f (r) %3D n! n=0 (In 2)" n! n=0 The interval of convergence of this power series is (-, +x).
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:Show detailed solution for all.
a) Provide the Maclaurin series expansion of f(x)
In (3 — 2л")
%3D
b) Find its interval of convergence
Example way of solving:
The function f (x) = 2" has a Maclaurin series expansion.
Solving for f(m) (0),
f(0) (2) = 2"
f(1) (x) = In 2 - 2"
f(2) (x) = (In 2)* - 2"
f(3) (x) = (In 2)3 . 2"
f(4) (x) = (In 2)* - 2"
f(0) (0)
= 1
f(1) (0) = In 2
f(?) (0) = (In 2)?
f(3) (0) = (In 2)*
f(4) (0) = (In 2)4
For n 2 0, f(n) (r) = (In 2)" - 2"
f(m) (0) = (In 2)"
So, the Maclaurin series expansion of f (x) = 2" is
f(n) (0),
too
f (r) =
n!
n=0
+oo
(In 2)"
n!
n=0
The interval of convergence of this power series is (-00, +o0).
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