Question

How many nonzero terms of the maclaurin series for ln(1 x) do you need to use to estimate ln(1. 4) to within 0. 0001?

Answer 1

We need at least 7 terms of the Maclaurin series for ln(1 + x)  to estimate ln 1.4 to within 0.0001

For given question,

We have been given a function f(x) = ln(1 + x)

We need to find  the estimate of In(1.4) within 0.001 by applying the function of the Maclaurin series for f(x) = In (1 + x)

The expansion of ln(1 + x) about zero is:

$ln(1+x)=x-\frac{x^2}{2} + \frac{x^3}{3} -\frac{x^4}{4} +\frac{x^5}{5} -\frac{x^6}{6} +.~.~.$

where -1 ≤ x ≤ 1

To estimate the value of In(1.4), let's replace x with 0.4

$\Rightarrow ln(1+0.4)=0.4-\frac{0.4^2}{2} + \frac{0.4^3}{3} -\frac{0.4^4}{4} +\frac{0.4^5}{5} -\frac{0.4^6}{6} +.~.~.$

From the above calculations, we will realize that the value of  $\frac{0.4^5}{5}=0.002048$ and $\frac{0.4^6}{6}=0.000683$  which are approximately equal to 0.001

Hence, the estimate of In(1.4) to the term $\frac{0.4^6}{6}$  is enough to justify our claim.

Therefore,  we need at least 7 terms of the Maclaurin series for function ln(1 + x)  to estimate ln 1.4 to within 0.0001

Learn more about the Maclaurin series here:

https://brainly.com/question/16523296

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