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 & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.5: The Binomial Theorem

Problem 44E

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