University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 9 - Section 9.10 - The Binomial Series and Applications of Taylor Series - Exercises - Page 549: 5



Work Step by Step

Formula to find the binomial series is: $(1+x)^m=1+\Sigma_{k=1}^\infty \dbinom{m}{k}x^k$ Here, $\dbinom{m}{k}=\dfrac{m(m-1)(m-2).....(m-k+1)}{k!}$ Now, $(1+\dfrac{x}{2})^{-2}=1-2(\dfrac{x}{2})(-2x)+\dfrac{(-2)(-3)(\dfrac{x}{2})^2}{2!}+\dfrac{(-2)(-3)(-4)(\dfrac{x}{2})^3}{3!}+...$ or, $=1-x+\dfrac{(-2)(-3)(\dfrac{1}{4})x^2}{2!}+\dfrac{(-2)(-3)(-4)(\dfrac{1}{8})x^3}{3!}+...$ Thus, the first four terms are: $1-x+\dfrac{3}{4}x^2-\dfrac{1}{2}x^3+.....$
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