Q14MP Find the Maclaurin series for th... [FREE SOLUTION]

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Chapter 1: Q14MP (page 1)

Find the Maclaurin series for the following functions $c o s [l n (1 + x)]$ :

Short Answer

Expert verified

Maclaurin series of $\cos [\ln (1 + x)]$ is $(\frac{x^{2}}{2} + \frac{x^{3}}{2} - \frac{5 x^{4}}{12} + 鈥� 鈥�)$ .

Step by step solution

Concept and formula used to find the Maclaurin series of the given function

The Maclaurin series of $c o s x$ is expressed as follows:

$c o s x = 1 - \frac{x^{2}}{2} + \frac{x^{4}}{24} + 鈥� 鈥� .$

The Maclaurin series of $l n (1 + x)$ is expressed as follows:

$l n (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} 鈥� 鈥� . .$

Calculation to find the Maclaurin series of the function cos [1n(1 + x)]:

Substitute $x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} 鈥� 鈥� .$ for $x$ in expansion of $\cos x$ as follows:

role="math" localid="1664264756169" $\cos [\ln (1 + x)] = 1 - \frac{{(x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} 鈥� 鈥�)}^{2}}{2} + \frac{{(x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} 鈥� 鈥�)}^{4}}{24} + 鈥� 鈥� \cos [\ln (1 + x)] = 1 - \frac{(- 576 x^{2} + 576 x^{3} - 480 x^{4})}{1152} + 鈥� 鈥� . \cos [\ln (1 + x)] = \frac{x^{2}}{2} + \frac{x^{3}}{2} - \frac{5 x^{4}}{12} + 鈥� 鈥� .$

Thus, the Maclaurin series of role="math" localid="1664264808184" $\cos [\ln (1 + x)]$ is $(\frac{x^{2}}{2} + \frac{x^{3}}{2} - \frac{5 x^{4}}{12} + 鈥� 鈥�)$ .

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91影视

Find the Maclaurin series for the following functions $c o s [l n (1 + x)]$ :

Short Answer

Step by step solution

Concept and formula used to find the Maclaurin series of the given function

Calculation to find the Maclaurin series of the function cos [1n(1 + x)]:

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91影视

Find the Maclaurin series for the following functions cos[ln(1 + x)]:

Short Answer

Step by step solution

Concept and formula used to find the Maclaurin series of the given function

Calculation to find the Maclaurin series of the function cos [1n(1 + x)]:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Fluids

Electricity and Magnetism

Energy Physics

Electromagnetism

Waves Physics

Study anywhere. Anytime. Across all devices.

Find the Maclaurin series for the following functions $c o s [l n (1 + x)]$ :