Problem

Find the first three nonzero terms of the Maclaur…

Problem 18 Easy Difficulty

Find the first three nonzero terms of the Maclaurin expansion of the given functions.
$$f(x)=\ln (1+4 x)^{2}$$

Answer

$8 x-16 x^{2}+\frac{128}{3} x^{3}$

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Calculus 2 / BC

Basic Technical Mathematics with Calculus

Chapter 30

Expansion of Functions in Series

Section 2

Mclaurin Series

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Video Transcript

All right. We want to find the first three non zero terms of the MacLaurin series expansion for F of X is equal to natural algorithm of one plus four X squared. This question is challenging our ability to find a series expansion of function F X. For this problem, we need to know what the MacLaurin series states. So MacLaurin series says that F x equals F zero plus zero times X plus double 5 0/2 factorial X squared plus 3 0/3 50 X cube. That is the term of the MacLaurin series and the threat of F zero divided by n factorial times X to the end, Allen of one plus four X squared zero is Ellen one or zero. So we have to find at least three to relations with non zero terms algorithms are as follows here, F one is two times 124 actually times 4/1 plus four X squared or 8/1 plus four x. This is +80 F two is negative 32/1 plus four X squared or negative 32. F three is 2 56/1 plus four X two or 2 36. So we have from F one F two and F three non zero terms plugging into MacLaurin series Kids, F x equals Ln one plus four X squared equals eight x minus 16 X squared plus 1, 28/3 X cubed and so on.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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