Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 10: Infinite Sequences and Series - Section 10.10 - The Binomial Series and Applications of Taylor Series - Exercises 10.10 - Page 633: 14



Work Step by Step

Apply the binomial series formula to determine the first four terms: $(1+x)^r=1+\Sigma_{k=1}^\infty \dbinom{r}{k}x^k$ Here, we have $\dbinom{r}{k}=\dfrac{r(r-1)(r-2).....(r-k+1)}{k!}$ $(1-\dfrac{x}{2})^{3}=1+(4)(-\dfrac{x}{2})+\dfrac{(4)(3)}{2!}(\dfrac{x^2}{4})+\dfrac{(4)(3)(2)}{3!}(\dfrac{-x^3}{8})+\dfrac{(4)(3)(2)(1)}{4!}(\dfrac{x^4}{16})=1-2x+\dfrac{3}{2}x^2-\dfrac{1}{8}x^3+\dfrac{1}{16}x^4$
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