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In the following exercises, compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of $f$ . $$f(x)=\ln (\cos x)$$
$$\ln (\cos x)=-\frac{1}{2 !} x^{2}-\frac{2}{4 !} x^{4}-\frac{4}{6 !} x^{6}$$
Calculus 2 / BC
Precalculus
Chapter 6
Power Series
Section 4
Working with Taylor Series
Powers and Polynomial
Johns Hopkins University
Oregon State University
University of Michigan - Ann Arbor
Idaho State University
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we want to find the 1st 3 turns of the MacLaurin series for Natural Log of Co signed X. Well, if we take the derivative well, let's call this f of X. We take the derivative of F of X. We get one over co sign X times, the derivative coast on its, which is minus sign X, and we get minus tangent tax. Now we know that the MacLaurin series for a tangent x is X plus 1/3 X cubed plus 2 15 x to the fifth and we're only talking wheeling in the 1st 3 terms from We're going to stop there. So the MacLaurin series for minus Tanja necks is just the negative of that minus X minus 1/3 X cubed minus 2 15 Sex to the fifth minus. And now, to find MacLaurin series for F of X, we just integrate turned by term. So yet minus X squared over too minus X to the fourth times for thirds. I'm sorry. Mine, Uh, times they're over 12 right? And then minus. It's gonna be a next to the sixth one divided by six. So too, over 90. And so on
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