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Problem 24 Medium Difficulty

Find a power series representation for $ f, $ and graph $ f $ and several partial sums $ s_n(x) $ on the same screen. What happens as $ n $ increases?
$ f(x) = \tan^{-1} (2x) $

Answer

$\sum_{n=0}^{\infty}(-1)^{n} \frac{2^{2 n+1} x^{2 n+1}}{2 n+1}$

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Video Transcript

So here is the power series for the tangent inverse of X function. So tangent inverse of ex uh is equal to x minus X cubed over three plus. Extra fifth over five minus x to the seventh over seven plus. An infinite more number of terms. Uh And this is the power series representation for tangent inverse affects as long as X is in the interval from negative 1 to 1. We want to find the power series representation for the uh Inverse tension of two x. And this is easily done. If tangent inverse of X equals this series, then tangent inverse of two X. Uh to find a series for this function simply substitute in two X everywhere you see X. So tangent inverse of two x Will be two x minus and then substitute two X and three X or two X. Two third over three plus two X to the 5th over five uh minus noticed plus uh And subtraction science keep alternating. So tangent inverse of two X uh minus two X to the seventh over seven plus. And of course the series goes on uh indefinitely. So now we just got to simplify uh two extra third, two extra fifth two x to the seventh. So we have two x minus law, two X to the third is really to to the third time's extra 32 to 38. So we have eight over three times execute plus and then two X to the fifth, two to the fifth is 32. So we'll have 32 X to the fifth over five. 32 X to the 5th Over five. And we can just put the 30 to 35 Uh -2 to the 7th. Double check in the calculator. Two to the seventh. Power is 1 28. So we're going to have 128 Times X to the 7th over seven plus. Obviously an infinite more number of terms. So this is the power series representation for detention inverse of two X. Now uh We can take partial sums of this series to approximate the tangent inverse of two x function. So as some with just one term would be two X. The some with two terms would be two x minus eight thirds. Execute if you wanted to use a some with three terms to approximate the function, then you just use the first three terms in the series. Um And likewise, if you want to use a some with four terms to approximate the tangent inverse of two X function. Use all four of these terms. So what we're going to do next is we're going to graft the tangent inverse of two X. Function on the graphing calculator. Using dez mo's. Then we're gonna grab some partial sums. Uh So let's use a some well graph uh Detainment inverse of two x function and then we'll graph partial stumps. So as to of ex uh simply means the partial some of this part of this power series using the first two terms. So we will graph two x minus eight thirds X cubed. And then uh we'll also graph Uh as three which is the partial sum using the first three terms of the series. So s three of x will be two X Uh -8/3 execute plus 32/5 extra to 5th. So we're gonna graph the actual function tangent inverse of two X. And then we're gonna graph uh the partial sum from its power series partial sum using the first two terms and a partial sum using the first three terms of the power series. And our goal is to compare or to see how well these partial sums. Uh These partial some functions approximate uh the tangent inverse function more specifically as we use more terms in from the power series. Uh Do we get a better approximation uh to the actual function? Okay, so here using dez most currently you see the graph of the red graph is the graph of the function tangent inverse of two. Xd actual function uh defined for excess between negative one and positive one. Now, since our power series that we derived for the tangent inverse of two X function uh really is only defined for X values between negative one and one. Uh Here is the first two terms from the power series. And when we graph it we're going to make sure that we keep X restricted to the interval from negative 1 to 1 and the same thing. When we take uh the third partial sum uh as three of X. Basically the power series, Using the first three terms. Once again, when we graph these three terms the first three terms of the power series, we're going to make sure that we keep X restricted between negative one And a positive one. So you're currently looking at this red graph is a graph of tangent inverse of two. Acts next I'm going to grab along with it. Uh Just um the partial some consisting of the first two terms of the power series, and I was actually actually pretty impressed just how close using just two terms from the power series approximates the actual graph. So take a look at this. Okay, um So the blue graph does a really good job of approximating the actual uh graph of the function in red. So the blue graph comes really close to the red graph at least for x values between negative one half and one half, it really kind of diverges away from it. Uh For excess between negative one and negative one half and one half Uh positive one. but for ex equal negative one half up to X equals positive one half. Using the partial some consisting of just two terms from the power series. Does a really good job of approximating the actual uh inverse tangent function at least on a portion of this interval. So now let's see how using the partial sum as three of ex uh some of the first three terms of the power series. Let's see how close uh three terms from the power series approximates. Uh The actual tangent universe function. Alright, this one does even a little bit better. Um So this green uh graph that you see here is using three terms from the power series. So you can see that three terms from the power series. Uh does a really nice job of approximating uh the inverse tangent function represented by the red graph. Once again, it starts to divert away from it a little bit when X is less than negative one half and when X is greater than one half, but it stays closer to it using three terms um stays closer to the actual inverse tangent function than to to term sticks. Now, let's look at all three of them together. All right, so the red graph is the actual inverse tangent function. Do Blue graph is when we use two terms from the power series, and the green graph is when we used three terms from the power series. So, the more terms you use from the power series, the closer you get to the actual function or the actual graph of the actual function. So the green graph does a lot better job than the blue graph does of staying close uh to the inverse tangent function

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