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Applications of Taylor Series We recall the definition of the general binomial coefficient (see Sheet 2): For any a Є R, n Є No, we define a-i+1 (²) = II²+1 For reference: The empty product is defined as one. Thus, (a) (a) Show that for f : R→ R with f(x) = (1+x)ª it holds that T[ƒ,0](x) = Σxo (2) xk by first deriving a general expression for f(n) (x). = 1. Hint: For the next part of the task, you may assume without proof that f(x) = T[ƒ,0](x) for |x| < 1. (b) We consider the functions g, h : (−1,1) → R with g(x)=√1+x and h(x) = 1 1+2 Determine, using part (a), the Taylor polynomial T5 [g, 0] (x) and the Taylor series T[h, 0](x). The general binomial coefficient must be calculated or simplified as much as possible in each case.

Applications of Taylor Series We recall the definition of the general binomial coefficient (see Sheet 2): For any a Є R, n Є No, we define a-i+1 (²) = II²+1 For reference: The empty product is defined as one. Thus, (a) (a) Show that for f : R→ R with f(x) = (1+x)ª it holds that T[ƒ,0](x) = Σxo (2) xk by first deriving a general expression for f(n) (x). = 1. Hint: For the next part of the task, you may assume without proof that f(x) = T[ƒ,0](x) for |x| < 1. (b) We consider the functions g, h : (−1,1) → R with g(x)=√1+x and h(x) = 1 1+2 Determine, using part (a), the Taylor polynomial T5 [g, 0] (x) and the Taylor series T[h, 0](x). The general binomial coefficient must be calculated or simplified as much as possible in each case.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Applications of Taylor Series We recall the definition of the general binomial coefficient (see Sheet 2): For any a Є R, n Є No, we define a-i+1 (²) = II²+1 For reference: The empty product is defined as one. Thus, (a) (a) Show that for f : R→ R with f(x) = (1+x)ª it holds that T[ƒ,0](x) = Σxo (2) xk by first deriving a general expression for f(n) (x). = 1. Hint: For the next part of the task, you may assume without proof that f(x) = T[ƒ,0](x) for |x| < 1. (b) We consider the functions g, h : (−1,1) → R with g(x)=√1+x and h(x) = 1 1+2 Determine, using part (a), the Taylor polynomial T5 [g, 0] (x) and the Taylor series T[h, 0](x). The general binomial coefficient must be calculated or simplified as much as possible in each case.
Transcribed Image Text:Applications of Taylor Series We recall the definition of the general binomial coefficient (see Sheet 2): For any a Є R, n Є No, we define a-i+1 (²) = II²+1 For reference: The empty product is defined as one. Thus, (a) (a) Show that for f : R→ R with f(x) = (1+x)ª it holds that T[ƒ,0](x) = Σxo (2) xk by first deriving a general expression for f(n) (x). = 1. Hint: For the next part of the task, you may assume without proof that f(x) = T[ƒ,0](x) for |x| < 1. (b) We consider the functions g, h : (−1,1) → R with g(x)=√1+x and h(x) = 1 1+2 Determine, using part (a), the Taylor polynomial T5 [g, 0] (x) and the Taylor series T[h, 0](x). The general binomial coefficient must be calculated or simplified as much as possible in each case.
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