How did we get the power series summation? How did we know to set it up like this.( Circled in pink)

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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How did we get the power series summation? How did we know to set it up like this.( Circled in pink)

Basic Skills »
9-26. Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero tern
centered at a.
b. Write the power series using summation notation.
C.
Determine the interval of convergence of the series.
1
9.
f(x) = -, a = 1
%3D
Transcribed Image Text:Basic Skills » 9-26. Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero tern centered at a. b. Write the power series using summation notation. C. Determine the interval of convergence of the series. 1 9. f(x) = -, a = 1 %3D
k!
k=0
k=0
11.3.9
a. f(x) =
1
so f(1) = 1. f'(x)
3 so f'(1) = -2. f"(x)
24
so f"(1) = 6. f"(x) = –
SO
%3D
%3D
%3D
f"(1) = -24. So the Taylor series is
15 SO
%3D
1- 2(x – 1) + 6
2!
(x-1)2
(x- 1)3
24
+.··=1– 2(x – 1) + 3(x – 1)² – 4(x – 1)3 + ..
3!
|
b. E(-1)*(k+1)(x – 1)*.
k=0
(-1)k+1(k + 2)(x – 1)k+1
(-1)k(k + 1)(x – 1)k
C. r= lim
= |x – 1|. This is less than 1 for -1 <x -1< 1, or 0< I< 2.
%3D
The series diverges at the endpoints by the Divergence Test, so the interval of convergence is (0, 2).
Copyright 2019 Pearson Education, Inc.
2.
Transcribed Image Text:k! k=0 k=0 11.3.9 a. f(x) = 1 so f(1) = 1. f'(x) 3 so f'(1) = -2. f"(x) 24 so f"(1) = 6. f"(x) = – SO %3D %3D %3D f"(1) = -24. So the Taylor series is 15 SO %3D 1- 2(x – 1) + 6 2! (x-1)2 (x- 1)3 24 +.··=1– 2(x – 1) + 3(x – 1)² – 4(x – 1)3 + .. 3! | b. E(-1)*(k+1)(x – 1)*. k=0 (-1)k+1(k + 2)(x – 1)k+1 (-1)k(k + 1)(x – 1)k C. r= lim = |x – 1|. This is less than 1 for -1 <x -1< 1, or 0< I< 2. %3D The series diverges at the endpoints by the Divergence Test, so the interval of convergence is (0, 2). Copyright 2019 Pearson Education, Inc. 2.
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