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A. Find the first four nonzero terms of the Taylor series centered at 0 for the given finction.b. Use the first four terms of the series to approximate the given qu...

Question

A. Find the first four nonzero terms of the Taylor series centered at 0 for the given finction.b. Use the first four terms of the series to approximate the given quantity.$$f(x)=(1+x)^{-2 / 3} ; ext { approximate } 1.18^{-2 / 3}$$

a. Find the first four nonzero terms of the Taylor series centered at 0 for the given finction. b. Use the first four terms of the series to approximate the given quantity. $$f(x)=(1+x)^{-2 / 3} ; \text { approximate } 1.18^{-2 / 3}$$



Answers

Taylor series and interval of convergence a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. $$f(x)=3^{x}, a=0$$

For this problem, we're gonna take a few derivatives off one over X Evaluated equals one. Find the 1st 4 non zero terms and its Taylor series. So we start with the function of one over X. This is the same as X to the negative one. If we evaluate this at one will get 1/1. Just want we take the first derivative of this function using the power rule. We're gonna negative X to the negative, too. Reevaluate this. That one really negative 1/1 squared, which is just one. You take the second derivative. We're going to get using the power rule two times X to the negative three, which was being finally won to and now the third derivatives of again Using the power rule will just be negative. Six x to the negative. Fourth this evaluated at one we'll just be negative. Six. You then plug this into the definition for the Taylor series, which starts with F of A, which is one plus f prime of A which is negative. One over one factorial times X minus A which is one plus to over two Factorial tens X minus, one squared minus six over three factorial X minus one cubed and so on. Now weaken simplifies in terms. So one minus 1/1 factorial is just one. So one minus X minus one to the first power plus 2/2 factorial. Two factorial issues too two divided by two is just one. I was just X minus one squared minus 6/3. Factorial is 6/3 times to three times to a six. So this is also just one minus X minus one to the third power and so on. Writing this generally well, give us something K equals zero to infinity. Uh, negative one to the cave powers we have start with a positive, An alternate between negative and positive. After that Times X minus one to the case power.

To start. We're gonna take a few derivatives of one over X, evaluate them at a equals two. So if we start with our function, be one over X. This is the same as X negative First power Evaluating this, a two will give to to the negative first power evaluate the first derivative. This will be negative X to the negative to power. Evaluating this at to be negative to to the negative too. Second, derivatives will be using the power rule to X to the negative Third power does. Evaluated to will be two times to to the first power times two in the negative third power which is to to the negative to power the third derivative it X will then be negative six times extra negative fourth power This evaluated it too. B negative six times x two negative fourth power were X is to We can then plug this into the definition from the Taylor series which starts with f of A, which is true in the negative one plus f prime of A which is negative 1/2 squared times one over two factorial times X minus a x minus two plus 1/2 squared times. 1/3 fact Meriel, Times X minus two All squared Apologies. The messed up with the fact Orioles, This has to be zero factorial. This is one factorial in this two factorial. Plus, we're just a minus six over to to the fourth times 1/3 factorial times X minus two to the third power and someone we can simplify this since the first term is two of the negative one and x minus 2 to 0 with power is just one minus one. Rewrite this as to the negative to power times X minus two to the first power plus to squared times two factorial two factorial is just too Services to the negative Third power times X minus two squared minus three. Factorial is three times two times one is able to six since these two were equivalent. So this is minus two to the negative. Fourth power comes explainers to the third power and so on. And now we see a pattern here so we can make a General Siri's represented representation for this which is we started positive one and go to negative one. So we need a negative one to the K Times X minus two to the chaos power over to to the K plus one power

To start here. We're gonna take some derivatives that the natural log of X and evaluate them in a equals three. So for so with the function being able to national log of X, evaluate this at a which is three. We're gonna get the natural order three, we take the first derivative of the natural log. This is one over X or extra negative one. Evaluate this. It three. We're just going to get three to the negative first power. If you take the second derivative of X, were using the power rule gonna get negative X to the negative to power. This evaluated three is going to be negative three to the negative to power. We take the third derivative of X using the power rule learning a two x two negative third power evaluating this at three running at two times three to the negative third power. Now we can use these values. Upload them into the definition of a tailor. Siri's. It starts with F of A, which is the natural order three plus f prime of A, which is three and negative one times X minus A where a is three plus the second derivative it a which is negative. Three to the negative, too. Times 1/2 factorial times X minus three squared plus 32 negative third power times two times one of the three sectorial times X minus three cubed and so and now we can rearrange in simplifies in terms to get this is natural log of three times three to the negative. One times 1/1 is just one. I was actually native third power raised to the first power minus reading the negatives. Negative, too times to factual Religious too. This is times 1/2 times X minus three. The second power class three sectorial is three times two times one. We can cancel out one factor too in three sectorial with this too. To get that, this is three to the negative. Three times 1/3 times X minus three you and so on. And now we can see a pattern in the terms after the first. So after the natural order three. You know that this is the natural order three plus the sum. Okay, go zero to infinity of your alternating positive and negative science when they would have won to the K times X minus three to the K plus one power divided by three to the K plus one power times, just K plus one.

For this problem, we're going to start by taking some derivatives of Evening X or evaluate them at a which is the natural lager too. So if we start with our function, which is either the ex, we evaluate this at a just natural log of to We'll get E. It's a natural lager, too, and one property of the exponential function is that you have e raised to the natural log of some number. A will always just be a in this case, are a is to so we'll just end up with two. We take the derivative of E to the X. It's just gonna give us back even the X. It's when we evaluate this at the natural lock up to this was the same Mr Function in national lager, too, which would just be, too. And we'll see for the second derivative yet again, we're gonna get you the X evaluating that at the National Aga too. Well, give us too. And finally the third derivative well again, you a seat of the X and evaluated the national Aga too. I would just be too. Now we can plug this into the definition for the tailor Siri's, which starts with f of age, were evaluated to be to plus after Prime a day which was two times X minus a where a is the national longer too, plus the second derivative a day which was two over two factorial times x minus the natural log But you squared plus to over three factorial times X minus the natural log to all cubed and so on. And these are first for non zero terms and we can see that we have a common factor of two in every term so we can factor that out to get that this is two times one. I'm gonna rewrite one as 1/1 factorial, since one factorial is one times x minus National August 2 raised to the zero with power. Anything raises your with our would be once in his first term is just one plus 1/1 factorial times X minus, natural log of to to the first power. Actually to keep the pattern going, one factorial is equal to zero factorial still equal the one plus 1/2 factorial X minus Natural lager, too. Squared this 1/3 factorial times X minus two natural are too. Oh, cute and so on. So now we can see inside of the brackets that we have a pattern in which we could make a serious representation of We have increasing energy powers of exercise the natural log of two. So we can have X minus the natural log of to raised to the cave power over just k factorial.


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