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Problem 28 Medium Difficulty

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {n!x^n}{1 \cdot 3 \cdot 5 \cdot \cdot \cdot \cdot \cdot (2n - 1)} $


$R=2 \quad$ interval of convergence: $(-2,2)$


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Video Transcript

we'LL figure out our radius of convergence without taking any shortcuts would just use the ratio test here. So lemon is n goes to infinity of a n plus one over and where a And is this whole chunk here, including the X values? Okay, so this might get a little messy. So this is a limit as n goes to infinity Absolute value X to the n plus one divided by X to the n is just x Okay, And then we still have this in plus one factorial. And we still have this whole chunk happening down here one times three times that that that times two and minus one times two and plus one minus one for the A N plus one guy, we have to go all the way until two times in plus one minus one and divided by alien is something as multiplying by the reciprocal of Anne. So now we're going to be multiplying by one times three times, dot, dot, dot all the way up to times two in minus one, and we're going to be divided by in factorial. Okay, so this chunk here is going to cancel out with this whole chunk here, the two times in plus one minus one is going to survive and in factorial is one times two times three times. Not that that times in, we're going to get everything within factorial to cancel out with the in plus one factorial but the n plus one in the in fact plus one, factorial will survive. So this is going to be simplified to limit as n goes to Infinity Act's times in plus one. Nothing's going to cancel out with the M plus one here, and then nothing will cancel out with this two times and plus one minus one. So that's still going to be there. Okay, as N goes to infinity in plus one divided by two times in plus one minus one, that's going to be one half because this is, ah, linear term with leading coefficient one. This is a linear term, with leading coefficient to sew as n goes to infinity, the limit is one divided by two, which is one half So this is absolute value of X over to, and when we're trying to figure out where we get convergence with the ratio tests, remember, we want for this to be strictly less than one. Okay, so that means that absolute value of exes. Lesson two, the radius of convergence. It's going to be too right cause X is going to be trapped between minus two and positive too. The length of that interval is for radius of convergence is half of the length of the interval of convergence. Okay, so now we need to figure out whether or not we include minus two and whether or not we include too. Okay, so what happens if we plug in X equals to here? X is equal to two than we have in factorial times to the end divided by one times. Three times that. That that times two and minus one case. Remember what X R what in Factorial is in? Factorial is one times two times three times that that that times in two to the end as two times two times that. That that times two where we have in different copies of two here. So if we multiply in factorial by two to the end, we can do one times two, two times two three times to each one of the factors in this product can be multiplied by one of these. Choose here. And then we can write this as one times two, which is two two temps to which is four, three times, too, which is six that that that all the way up to to end. And we're dividing by one times three times. Doctor. At that. Yes. I should put a five here. So we have the dot that thoughts in the same place times two in minus one. All right. But now we just put parentheses in the right place. Then we can see that this is not going to converge because the terms are not even going to go to zero. So we just put Prentice's around each of the guys here, apprentices around to over one footprint of seas around former three that that that keep going until we have Prentiss is around two in over two and minus one. And notice that each one of these terms is bigger than one. So this whole thing has to certainly be bigger than one. So in particular, it can't possibly go to zero. So these terms do not go to zero. If the terms don't go to zero and we can't possibly get convergence. So we get divergence and we get the same type of thing happening. If we plug in X equals minus to plug in X equals minus two, we get the same type of thing. Except now we would have AA minus one to the inn happening here. But again, it's not going to change the fact that our terms are not going to zero. Okay, so we get divergence if we plug in two or minus two into R sum here. So both those values we have to throw out of our interval of convergence. So we leave that as an open interval here. So not included minus two and not including positive, too.

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