Chapter 1: Q35p (page 32)

$\frac{x}{s i n x}$

Short Answer

Expert verified

$\frac{x}{\sin (x)} = 1 + \frac{x^{2}}{6} + \frac{7 x^{4}}{360} + \dots$

Step by step solution

Given information

The Maclaurin expansion of the function $\frac{x}{\sin x}$ is to be evaluated.

Definition of Maclaurin series

The definition of the Maclaurin series is defined.

$s i n (x) = x - \frac{x^{3}}{6} + \frac{x^{5}}{120} + \dots$

$(1 + x)^{- 1} = 1 - x + x^{2} - x^{3} + \dots$

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

State the Maclaurin series and evaluate.

$\sin (x) = x - \frac{x^{3}}{6} + \frac{x^{5}}{120} + \dots$

$(1 + x)^{- 1} = 1 - x + x^{2} - x^{3} + \dots$

Part-B Step 3: Use the series to create an expression

Use the Maclaurin series to create the expression.

$\frac{x}{\sin (x)} = \frac{1}{\frac{\sin (x)}{x}}$

$\frac{\sin (x)}{x} = \frac{x - \frac{x^{3}}{6} + \frac{x^{5}}{120} + \dots}{x}$

$= 1 - \frac{x^{2}}{6} + \frac{x^{4}}{120} + \dots$

Continue to evaluate.

$\frac{x}{\sin (x)} = \frac{1}{\frac{\sin (x)}{x}}$

$= \frac{1}{1 - \frac{x^{2}}{6} + \frac{x^{4}}{120} + \dots} = {(1 + [- \frac{x^{2}}{6} + \frac{x^{4}}{120} + \dots])}^{- 1}$

Part-C Step 3: Use the Maclaurin series to evaluate

Replace $x$ with $[- \frac{x^{2}}{6} + \frac{x^{4}}{120} + \dots]$ and evaluate the Maclaurin series.

${(1 + [- \frac{x^{2}}{6} + \frac{x^{4}}{120} + \dots])}^{- 1} = 1 - [- \frac{x^{2}}{6} + \frac{x^{4}}{120} + \dots] + {[- \frac{x^{2}}{6} + \dots]}^{2} + \dots$

$\frac{x}{\sin (x)} = {(1 + [- \frac{x^{2}}{6} + \frac{x^{4}}{120} + \dots])}^{- 1}$

$\frac{x}{\sin (x)} = 1 + \frac{x^{2}}{6} + \frac{7 x^{4}}{360} + \dots$

Thus, the final answer is $\frac{x}{\sin (x)} = 1 + \frac{x^{2}}{6} + \frac{7 x^{4}}{360} + \dots$

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$\frac{x}{s i n x}$

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 the series to create an expression

Part-C Step 3: Use the Maclaurin series to evaluate

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xsinx

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 the series to create an expression

Part-C Step 3: Use the 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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$\frac{x}{s i n x}$