Chapter 1: Q31p (page 32)

$c o s (e^{x} - 1)$

Short Answer

Expert verified

$\cos (e^{x} - 1) = 1 - \frac{x^{2}}{2} - \frac{x^{3}}{2} - \frac{x^{4}}{4} + \dots$

Step by step solution

Given information

The Maclaurin expansion of the function $\cos (e^{x} - 1)$ is to be evaluated.

Definition of Maclaurin series

The definition of the Maclaurin series is defined.

$e^{x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \dots$

$c o s (e^{x} - 1) = 1 - \frac{x^{2}}{2} - \frac{x^{3}}{2} - \frac{x^{4}}{4} + \dots$

e^{x} - 1 = x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \dots

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

State the Maclaurin series.

$e^{x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \dots$

$\cos (e^{x} - 1) = 1 - \frac{x^{2}}{2} - \frac{x^{3}}{2} - \frac{x^{4}}{4} + \dots$

$e^{x} - 1 = x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \dots$

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

Replace $x$ with $[x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \dots]$ in the second equation.

$\cos (e^{x} - 1) = 1 - \frac{{[x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \dots]}^{2}}{2} + \frac{{[x + \frac{x^{2}}{2} + \dots]}^{4}}{24} + \dots$

$\cos (e^{x} - 1) = 1 + \frac{(- 576 x^{2} - 576 x^{3} - 288 x^{4})}{1152} + \dots$

$\cos (e^{x} - 1) = 1 - \frac{x^{2}}{2} - \frac{x^{3}}{2} - \frac{x^{4}}{4} + \dots$

Thus, the final answer is $\cos (e^{x} - 1) = 1 - \frac{x^{2}}{2} - \frac{x^{3}}{2} - \frac{x^{4}}{4} + \dots$

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$c o s (e^{x} - 1)$

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 Maclaurin series to evaluate

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cos(ex-1)

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 Maclaurin series to evaluate

One App. One Place for Learning.

Most popular questions from this chapter

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$c o s (e^{x} - 1)$