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Problem(16pts) . Find the first four nonzero terms of the Maclaurin series (by any method) for f(x) = Then find the interval of convergence of this series. (1 + 3...

Question

Problem(16pts) . Find the first four nonzero terms of the Maclaurin series (by any method) for f(x) = Then find the interval of convergence of this series. (1 + 31)2

Problem (16pts) . Find the first four nonzero terms of the Maclaurin series (by any method) for f(x) = Then find the interval of convergence of this series. (1 + 31)2



Answers

Find the Maclaurin series for $$f(x)=\ln \frac{1+x}{1-x}$$ and determine its radius of convergence. Use the first four terms of the series to approximate $\ln 3$.

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For this problem we are asked to find them. Excuse me. The MacLaurin polynomial is of orders and equal zero through four. And then find the MacLaurin series for the function 1/1 plus X. In sigma notation. So at n equals zero. We'll just have a constant value of one at n equals one. We will get one minus X. At n equals two. We'll have one minus X plus X squared at n equals three of one minus X plus x squared minus X cubed. And at n equals four. We'll have one minus X plus X squared minus x cubed X cubed plus X the power of four. So now we can see the pattern rather clearly. We can write our sigma notation form of the MacLaurin series as the sun from M equals zero up to n. Of Well now we'll note the odd power of X terms have negative signs out front, whereas the evens are positive. So we can put a negative one to the power of em out front. Then we have just X to the power of them giving us our MacLaurin series. All right,

Problem. 30 night with this problem just calculates foreign serious about e to that square and just times after the force. That's okay. So you two have square his ego's too X square plus Oh, there's one plus X Square this extra floors over, too. So just times after the forest, it's time for. So he sacks the forest bus, exit the stakes and plus after the aid over, too, that the first term's first returns and we want to find for with the serious covered is absolutely. You know, this convergence when next in our So the whole terms just covered that they are. I wanna see ability to be pretty.


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