Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that R₂(x) → 0.] f(x) = 10x - 2x³, a = -1

Algebra and Trigonometry (6th Edition)
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.]
f(x) = 10x - 2x³,
a = -1
∞
R =
n = 0
∞
n = 0
*Wi *Wi
n = 0
n = 0
∞
n = 0
f(n)(−1) (x + 1)^
n!
= −8 + 4(x + 1) + 6(x + 1)² .
f(n)(−1)(x + 1)^ = −8 + 4(x + 1) + 2(x + 1)² .
n!
f(n)(−1)(x + 1)^
n!
2(x + 1)³
f(n)(−1) (x + 1)″ = −8 − 4(x + 1) + 6(x + 1)² + 2(x + 1)³
n!
6(x + 1)³
Find the associated radius of convergence R.
= −8 + 6(x + 1) + 4(x + 1)² − 2(x + 1)³
f(n)(−1) (x + 1)″ = −8 − 4(x + 1) + 2(x + 1)² + 6(x + 1)³
n!
Transcribed Image Text:Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] f(x) = 10x - 2x³, a = -1 ∞ R = n = 0 ∞ n = 0 *Wi *Wi n = 0 n = 0 ∞ n = 0 f(n)(−1) (x + 1)^ n! = −8 + 4(x + 1) + 6(x + 1)² . f(n)(−1)(x + 1)^ = −8 + 4(x + 1) + 2(x + 1)² . n! f(n)(−1)(x + 1)^ n! 2(x + 1)³ f(n)(−1) (x + 1)″ = −8 − 4(x + 1) + 6(x + 1)² + 2(x + 1)³ n! 6(x + 1)³ Find the associated radius of convergence R. = −8 + 6(x + 1) + 4(x + 1)² − 2(x + 1)³ f(n)(−1) (x + 1)″ = −8 − 4(x + 1) + 2(x + 1)² + 6(x + 1)³ n!
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