Chapter 1: Q30p (page 32)

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

Short Answer

Expert verified

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

Step by step solution

Given information

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

Definition of Maclaurin series

The definition of the Maclaurin series is defined.

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

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

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

State the Maclaurin series.

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

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

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

Use both Maclaurin series to evaluate.

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

$= [1 - \frac{x}{2} - \frac{x^{2}}{8} - \frac{x^{3}}{16} + \dots] \times [1 - \frac{x}{2} + \frac{3 x^{2}}{8} - \frac{5 x^{3}}{16} + \dots]$

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

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

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

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

Short Answer

Step by step solution

Given information

Definition of Maclaurin series

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

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

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1-x1+x

Short Answer

Step by step solution

Given information

Definition of Maclaurin series

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

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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Medical Physics

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Engineering Physics

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Study anywhere. Anytime. Across all devices.

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