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Show detailed solution for all. a) Provide the Maclaurin series expansion of f (x) In(3 - 2.x) b) Find its interval of convergence

Show detailed solution for all. a) Provide the Maclaurin series expansion of f (x) In(3 - 2.x) b) Find its interval of convergence

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 72E
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Give the Maclaurin series of expansion and its interval of convergence (include the checking of values)

Show detailed solution for all. a) Provide the Maclaurin series expansion of f (x) In(3 – 2.x) b) Find its interval of convergence Example way of solving: The functionf (r) 2" has a Maclaurin series expansion. Solving for f(n) (0), f(0) (x) = 2" f(1) (r) = In 2 2" f(2) (r) = (In 2)2 - 2" f(3) (2) = (In 2)3 . 2" f(4) (x) = (In 2)* - 2" f(0 (0) = 1 f(1) (0) = In 2 f(2) (0) = (In 2)2 f(3) (0) = (In 2)3 f(4) (0) = (In 2)4 For n 2 0, f(m) (x) = (In 2)" 2" f(n) (0) = (In 2)" %3D So, the Maclaurin series expansion of f (x) = 2" is +oo f(m) (0) f (x) = E n! %3D n=0 +0o (In 2)" n! n=0 The interval of convergence of this power series is 00, +0).
Transcribed Image Text:Show detailed solution for all. a) Provide the Maclaurin series expansion of f (x) In(3 – 2.x) b) Find its interval of convergence Example way of solving: The functionf (r) 2" has a Maclaurin series expansion. Solving for f(n) (0), f(0) (x) = 2" f(1) (r) = In 2 2" f(2) (r) = (In 2)2 - 2" f(3) (2) = (In 2)3 . 2" f(4) (x) = (In 2)* - 2" f(0 (0) = 1 f(1) (0) = In 2 f(2) (0) = (In 2)2 f(3) (0) = (In 2)3 f(4) (0) = (In 2)4 For n 2 0, f(m) (x) = (In 2)" 2" f(n) (0) = (In 2)" %3D So, the Maclaurin series expansion of f (x) = 2" is +oo f(m) (0) f (x) = E n! %3D n=0 +0o (In 2)" n! n=0 The interval of convergence of this power series is 00, +0).
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