Hi. Well, I have a problem with this one. It asks me to approximate [tex]\ln(0.7)[/tex] using MacLaurin polynomial of fourth degree. And estimate the error.(adsbygoogle = window.adsbygoogle || []).push({});

So I used:

[tex]f(x)=\ln(x+1)[/tex] [tex]f'(x)=\displaystyle\frac{1}{1+x}[/tex] [tex]f''(x)=\displaystyle\frac{-1}{(1+x)^2}[/tex] [tex]f'''(x)=\displaystyle\frac{2}{(1+x)^3}[/tex] [tex]f^4(x)=\displaystyle\frac{-6}{(1+x)^4}[/tex] [tex]f^5(x)=\displaystyle\frac{24}{(1+x)^5}[/tex]

And the MacLaurin polynomial I got:

[tex]P_4(x)=x-\displaystyle\frac{x^2}{2}+\displaystyle\frac{x^3}{3}-\displaystyle\frac{x^4}{4}[/tex]

So then

[tex]P_4(-0.3)=-0.3-\displaystyle\frac{0.3^2}{2}-\displaystyle\frac{0.3^3}{3}-\displaystyle\frac{0.3^4}{4}=-0.356025[/tex]

Here is the problem, when calculating the error I got:

[tex]R_5(-0.3)=\displaystyle\frac{-0.3^5}{5(1+\alpha)^5}[/tex]

Where [tex]-0.3<\alpha<0[/tex]

So if I use [tex]\alpha=0[/tex] I should get an upper boundary for the error, but [tex]\displaystyle\frac{-03^5}{5}{\leq{\epsilon}\Rightarrow{-0.000486{\leq{\epsilon}}[/tex]

But with wolframalpha: http://www.wolframalpha.com/input/?i=ln(0.7)+0.356025

So the error is wrong, but I can't find where is my mistake. It should give something like 0.0006 where I got 0.0004.

Bye there.

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# Homework Help: Approximating ln(0.7) using MacLaurin formula

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