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Sagot :
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:
[tex]ln(1+x)=x-\frac{x^2}{2} + \frac{x^3}{3} -\frac{x^4}{4} +\frac{x^5}{5} -\frac{x^6}{6} +.~.~.[/tex]
where -1 ≤ x ≤ 1
To estimate the value of In(1.4), let's replace x with 0.4
[tex]\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} +.~.~.[/tex]
From the above calculations, we will realize that the value of [tex]\frac{0.4^5}{5}=0.002048[/tex] and [tex]\frac{0.4^6}{6}=0.000683[/tex] which are approximately equal to 0.001
Hence, the estimate of In(1.4) to the term [tex]\frac{0.4^6}{6}[/tex] 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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