Chapter 1: Q33p (page 32)

$\frac{1 - s i n x}{1 - x}$

Short Answer

Expert verified

$\frac{1 - \sin (x)}{1 - x} = 1 + \frac{x^{3}}{6} + \frac{x^{4}}{6} + \frac{19 x^{5}}{120} + \dots$

Step by step solution

Given information

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

Definition of Maclaurin series

The definition of the Maclaurin series is defined.

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

$(1 - x)^{- 1} = \sum_{n = 0}^{\infty} (\begin{matrix} - 1 \\ n \end{matrix}) (- 1)^{n} x^{n}$

$= 1 + x + x^{2} + x^{3} + x^{4} + x^{5} + \dots$

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

State the Maclaurin series.

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

$(1 - x)^{- 1} = \sum_{n = 0}^{\infty} (\begin{matrix} - 1 \\ n \end{matrix}) (- 1)^{n} x^{n}$

$= 1 + x + x^{2} + x^{3} + x^{4} + x^{5} + \dots$

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

Use the first Maclaurin series and evaluate.

$1 - \sin (x) = 1 - [x - \frac{x^{3}}{6} + \frac{x^{5}}{120} - \dots]$

$1 - \sin (x) = 1 - x + \frac{x^{3}}{6} - \frac{x^{5}}{120} + \dots$

Part-C Step 3: Evaluate the denominator and combine to find the answer

Use the second Maclaurin series to find the expansion and evaluate the find the answer.

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

$= [1 - x + \frac{x^{3}}{6} - \frac{x^{5}}{120} + \dots] \times [1 + x + x^{2} + x^{3} + x^{4} + x^{5} + \dots]$

$= [\begin{array}{l} [1 + x + x^{2} + x^{3} + x^{4} + x^{5} + \dots] - x \cdot [1 + x + x^{2} + x^{3} + x^{4} + x^{5} + \dots] + \\ \frac{x^{3}}{6} \cdot [1 + x + x^{2} + x^{3} + x^{4} + x^{5} + \dots] - \frac{x^{5}}{120} \cdot [1 + x + x^{2} + x^{3} + x^{4} + x^{5} + \dots] \end{array}]$

$\frac{1 - \sin (x)}{1 - x} = 1 + \frac{x^{3}}{6} + \frac{x^{4}}{6} + \frac{19 x^{5}}{120} + \dots$

Thus, the final answer is $\frac{1 - \sin (x)}{1 - x} = 1 + \frac{x^{3}}{6} + \frac{x^{4}}{6} + \frac{19 x^{5}}{120} + \dots$

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$\frac{1 - s i n x}{1 - x}$

Short Answer

Step by step solution

Given information

Definition of Maclaurin series

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

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

Part-C Step 3: Evaluate the denominator and combine to find the answer

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1-sinx1-x

Short Answer

Step by step solution

Given information

Definition of Maclaurin series

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

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

Part-C Step 3: Evaluate the denominator and combine to find the answer

One App. One Place for Learning.

Most popular questions from this chapter

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$\frac{1 - s i n x}{1 - x}$