Question
Inktegrate Pomer scties QuestionWrite out the first four non-zero terms of the power series rep esentation for f(z) In(4 41) by integrating the power series for f' Express your answer J5 sum_Provide your answer bclow:
Inktegrate Pomer scties Question Write out the first four non-zero terms of the power series rep esentation for f(z) In(4 41) by integrating the power series for f' Express your answer J5 sum_ Provide your answer bclow:
Answers
a. Find the first four nonzero terms of the binomial series centered at 0 for the given function. b. Use the first four nonzero terms of the series to approximate the given quantity. $$f(x)=\sqrt{1+x} ; \text { approximate } \sqrt{1.06}$$
So this problem really need to find the first four nuns returns MacLaurin series by power series representation and in general of convergence. So first thing we have is that this is going to be very similar to central log house. Next plus one. Since that's equal to X. I guess x squared over two. That's next to 3rd over three And 6 to the 4th floor. Main difference is that we take the derivative of this. Sure, we get Went over 1, 2 3 times. Everything else that will have afterwards. So, the main difference is that we multiply all of this By one of Round three. So we have as the first time it's ex lover Allen three minus X squared to Allen of three. That's next to the 3rd over 303, 64th Over four. And then a three. Okay, so that that gives us the summation of So next to the K over talented enough. Three temps. Okay, down here. And richest. Missings from chemicals want to infinity. Yeah, -1 to the K plus one here. All right. That's the full summation there and then want to get the interval of convergence. So, we have the limit as N approaches infinity. Next to the Plus one. All right. This one. So in the three times And times all in a three older. Next Yeah. Sympathised. two X to the end. I mean just X. Just one. So we have from negative 1 to 1 here. So let's check that out here. So add So, I think I have one goes to You don't want to to keep this one. Yeah. Which is always going to be negative sometimes, so therefore this too gracious. And on the other hand, we have one times negative, Aunt Swan over Kaelin of three. That coverage is by alternating serious. So therefore the interval a convergence is from negative 1-1, so.
To find the Mac chlorine series of this expression, we can use this formula and we substitute X with X squared. So their cycles ax squared miners ax squared, and this squired over to last square and then kills or three miners X square in central force or were full in accent. This equals ax squared miners X. To a force over to Plus extra 6/3 Minors. Act to the AIDS over four and exactly.
Find the binomial series, that 1st 4 terms for this. So here we have That P is equal to -3. And so then with that in mind here we're going to have Use the Formula one Plus PX plus p times p minus one X squared over two Plus P. Times P -1. The P -2. Next to 3rd over three factorial. So let's give us the first four terms. It's plug in. We have 1 -3 x plus that's negative three times negative four which is 12. That's 12 X squared over two. And then plus that would be 12 times thanks to five Next to the 3rd over six. So with that in mind here, you know that that equals one, weighs three X Plus four x squared minus. That's 60/6 which is minus 10 X. To the third. So this would be the first four terms we're looking for here and then from there. So we want approximately 1.1. Next uh 1.12 3 the bottom there. And so that's equal to on our one plus X to the third. That means taxes equal 2.1. So we're plugging in .1 in for any X values up there. And then as we're doing that here, so the value that we would get as we approximate it, it's going to be equivalent to 0.75
We can write this expression as one miner's extra force and then to a powerful house. Hence we can use this formula with an eco's half and we substitute X with minors. Extra forms has the echoes one plus a half times miners. Extra falls class. Uh huh harms minors. How over to times miners? Extra flaws squared plus mm hmm. Miners have minor three has overseas times minus X. To force killed and etcetera. They see girls. one miners half gets to a force miners one AIDS. Yeah. Acts to the 8th and minors. One over 16 times. Acts two trails and accept.