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Chapter 1: Q30p (page 32)

$\sqrt{\frac{1 - x}{1 + x}}$

Short Answer

Expert verified

$\sqrt{\frac{1 - x}{1 + x}} = 1 - x + \frac{x^{2}}{2} - \frac{x^{3}}{2} + 鈥�$

Step by step solution

01

Given information

The Maclaurin expansion of the function $\sqrt{\frac{1 - x}{1 + x}}$ is to be evaluated.

02

Definition of Maclaurin series

The definition of the Maclaurin series is defined.

$(1 - x)^{1 / 2} = 1 - \frac{x}{2} - \frac{x^{2}}{8} - \frac{x^{3}}{16} + 鈥�$

$(1 + x)^{- 1 / 2} = 1 - \frac{x}{2} + \frac{3 x^{2}}{8} - \frac{5 x^{3}}{16} + 鈥�$

03

Part-A Step 3: Begin by stating the Maclaurin series

State the Maclaurin series.

$(1 - x)^{1 / 2} = 1 - \frac{x}{2} - \frac{x^{2}}{8} - \frac{x^{3}}{16} + 鈥�$

$(1 + x)^{- 1 / 2} = 1 - \frac{x}{2} + \frac{3 x^{2}}{8} - \frac{5 x^{3}}{16} + 鈥�$

04

Part-B Step 3: Use both Maclaurin series to evaluate

Use both Maclaurin series to evaluate.

$\sqrt{\frac{1 - x}{1 + x}} = (1 - x)^{1 / 2} 脳 (1 + x)^{- 1 / 2}$

$= [1 - \frac{x}{2} - \frac{x^{2}}{8} - \frac{x^{3}}{16} + 鈥�] 脳 [1 - \frac{x}{2} + \frac{3 x^{2}}{8} - \frac{5 x^{3}}{16} + 鈥�]$

$\sqrt{\frac{1 - x}{1 + x}} = 1 + \frac{- 256 x + 128 x^{2} - 128 x^{3} + 鈥�}{256}$

$\sqrt{\frac{1 - x}{1 + x}} = 1 - x + \frac{x^{2}}{2} - \frac{x^{3}}{2} + 鈥�$

Thus, the final answer is $\sqrt{\frac{1 - x}{1 + x}} = 1 - x + \frac{x^{2}}{2} - \frac{x^{3}}{2} + 鈥�$

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Most popular questions from this chapter

$f (x) = s i n 鈥� x, 鈥� a = 蟺 / 2$

Find the sum of each of the following series by recognizing it as the Maclaurin series for a function evaluated at a point ${鈭�}_{n - 1}^{鈭�} \frac{2^{n}}{n!}$ .

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

$\lim_{x 鈫� 0} \frac{1 - e^{x^{3}}}{x^{3}}$ .

$\frac{1 + x}{1 - x}$

Question: Test the following series for convergence

${鈭�}_{n = 1}^{鈭�} \frac{(- 1)^{n} \sqrt{10 n}}{n + 2}$

See all solutions

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