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 $(\frac{x^{2}}{2} + \frac{x^{3}}{2} - \frac{5 x^{4}}{12} + \dots \dots)$ .

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:

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

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

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

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

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

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

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

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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)]:

One App. One Place for Learning.

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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

Further Mechanics and Thermal Physics

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Study anywhere. Anytime. Across all devices.

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