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Part B write the first four write the first four nonzero terms of the Maclaurin series for F prime the derivative of F express F prime as a rational function for the absolute value of x being less than R our radius of convergence so if we want to find f prime we could just take each of we could just take the derivative of each of these terms with respect to X and so we could just say if this is F then F prime and I'm not gonna scroll down just so I can see this up here we could just say F prime F prime of X is going to be gyu or the Maclaurin series for F prime of X maybe I should write it that way so let me write it this way Maclaurin Maclaurin Maclaurin it looked funny Maclaurin series for F prime F prime of X well it's going to be the sum from N equals 1 to infinity and we would just take the derivative of this right over here with respect to X and so this is just a application of the power rule take the exponent right over here multiply it by the coefficient so if you take n times this that cancels out with this n so it's going to be negative 3 to the N minus 1 and then decrement your exponent times X to the N minus 1 and so they want to write the first four nonzero terms of the Maclaurin series so that is going to be equal to so I'll write approximately equal to because we're all going to write the first four of terms of this infinite series and it should be clear what I did I just did the power rule here I looked at this exponent which is n multiplied by this coefficient which had an N in the denominator so that N and this n cancel out so I'm just left with negative 3 to the n minus 1 and then I decrement that exponent that's straight out of the power rule one of the first things you learned about taking derivatives and so if we want the first four nonzero terms when N equals one this is going to be [ __ ] three to the one minus one power let me just write it negative three to the one minus one times X to the one minus one that's when N equals one plus negative three negative three to the two minus one two minus one times X to the 2 minus one and then and actually I could well I could have written this as negative three X to the N minus one actually let me let me do that just for so this is this I could just write this they have the same exponent this will simplify a little bit as negative three X to the n minus one and so this is going to be approximately when n is equal to one this is going to be equal to zero so negative three X to the zeroth power is just going to be one when n is equal to when n is equal to two this is going to be two minus one so it's going to be the first power so negative three x to the first power so I can just write this as negative three X and then when n is three well this is going to be negative three x squared so negative three x squared is going to be negative three squared is nine x squared and then the fourth term is going to be negative three X to the four minus one power so to the third power so negative three to the third power is negative 27 times X to the third power so there you have it that's the first four nonzero terms of the Maclaurin series you could have also just looked over here and said okay the derivative of X is with respect to X is one derivative of negative three half x squared with respect to X is negative three you could have said the derivative of this is nine x squared right over there and then you would have had to write out the fourth term and take out this the derivative in the same way and you would have gotten this right over here so we did the first part we wrote the first four nonzero terms of the Maclaurin series for F Prime the derivative of F and then they say express F prime is a rational function for the absolute value of x being less than R so this sum this sum if we assume it convert we know the radius of convergence already so or assuming that we're dealing with X that are within the radius of convergence so this right over here you might recognize this so I could write it like that or I could also write it I could also write it if I take if I start at N equals zero so I could also write this as from N equals 1 to infinity of or actually from N equals zero to infinity of negative three X to the n minus one either I'm sorry to the N because now the first term was to the zero power that first term was the zero power so whether you do 1 minus 1 is where you start or you just start at 0 these two things are equivalent you might recognize these as a geometric series with common ratio of negative of negative 3x and so what's the sum of a geometric series with common ratio with a certain common ratio well it's going to be equal to the first term and regardless of how you view this the first term is going to be 1 divided by 1 minus the common ratio so our common ratio is negative 3x 1 minus negative 3x well that's just going to be 1 plus 3x if what I just did here looks unfamiliar to you I encourage you to watch the sum of infinite geometric series and not only do we show you this formula and how to apply it but we show how you can prove this format it's actually a pretty fun proof but anyway regardless of how you view this Maclaurin series it is an infinite geometric series and this assuming that our X is in our radius of convergence this is what our sum is this is what we are going to converge to
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