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Problem 20 Medium Difficulty

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

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

Answer

Interval of Convergence: $[0,1]$
Radius of Convergence: $R=\frac{1}{2}$

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Video Transcript

All right, We've got a question here. We've got the series. Fairly two equals one. We have two X minus one to the end. Power five, the squirrel and power. All right. Now here. Were asked to do is to find the radius of convergence and the interval of convergence of this series. All right, so from here, what we would do is we would first set our base of ends. You know, that's going to be equal to two X minus one to the end. Power over five to the end. Power square that. Yeah, this is possibly a This is where I'm wrong. Should be five to the end. Okay, That's r A F n and then are a of n plus one equal to the same thing. But we're just gonna substitute r n were n plus one five to the end plus one. And finally the square root of that possible. All right, So, in order to solve for the radiance of convergence what we do as we take the limit as n approaches infinity and we take the absolute value of a sub n plus one live to the end plus one and plus one divided by you're a sub n, which is the same thing as two X minus one. The end power over five to the end. Power squatted down. All right. And then when you go through and actually saw about this entire limit here, you'll get an answer that looks like one of five two multiplied by the square of two X minus one. Okay, now, once you get that value, what you're gonna do is you're gonna use the ratio test. You're going to see whether or not this is less than one. She would take your two X minus one here, 45 and you're going to determine whether or not it's less than one. First, you just multiply five. So excuse me. First you go through and solve this inequality. We can do that by multiplying five on both sides. Yet two X minus one is less than five, and then we'll get Excuse me. It should be an absolute value for that time. Include that. And so since it's an absolute value, you would also have a a negative five that it's greater than because it could be The X can be negative or positive. Then we have a less than a positive positive. Okay, here's what we're gonna do is we're gonna We can then eliminate our our absolute value because we've included a less greater than negative five and less than five. What we're gonna do is we're gonna add one on both sides, and then we'll divide by two. So we'll get our X to be anywhere from negative 2 to 3. Um, not including negative 2 93. Okay, so now that we've got our intervals, what we're gonna do is check to see if the end points are to be included. Um, what we would do is we would plug in negative, too, or our ex and we would solve. And we would plug in. Let's see, what's our series looking like here. You've got a two X minus one. We're setting a expected tonight too. You got that to the negative to power. Oh, I'm sorry. This is We're substituting X, not n still going to keep my aunts to bees and okay, and then we'll have a negative five to the end. Power live to the end square would end. And we know that our five to the end uh, negative five to the end and then r n our five to the end can be simplified and we'll get a negative one to the end. Power over five. Excuse me, route and all Right now we want to determine if it's convergent or not. We would set the limit. Four hour part B. Here. Excuse me, are part B seven You know, our combination here would be are a multiplied Barbie A would be a negative one to the end. Power your be some men, you go to one over square And so what you do is you take your limit as n approaches infinity or your B seven, which is one root end, and you end up getting a one over one over infinity, which is the same thing that zero. We know that if it's equal to zero by the alternating series test, that's what we're using here. By the way, you can see that it is in fact, convergent because it comes to a value You could say it is convergent Thank you where we can now check the endpoint when X is equal to three. So we would have to It's three. Excuse me to the end five to the end. The end. So we have five to the end. Live to the end. Just be one over n And here we took the peat P minus test. You know, we would say that wall. This can be written as one over end to the half, and we know that half is less than one. Therefore, the series is divergent. It's a half is less than 15 No p minus test for diverges. All right. And then finally, when we write out our interval of convergence, we know it doesn't include it does include negative, too. So we would put this solid bracket, and we know it doesn't include three. All right, And then the radiance of convergence can be found by taking a our ratio test. And we know that that is when your, um our you know, when that occurs, when this here is less than one, she would write out the two. Excuse me. Let's write that out again. So we have these values here. We have two X minus one is less than five, and then we have a a two x minus. One is less than five. You divide both sides by two. So if we just basically we just take this part here to determine the rate of radius and convergence, you'd say two X minus one less than five X minus half. It's less than five half, and you would then say so. This is our standard form here, X minus half, and whatever it is less than is what we write as our radius of convergence. Okay, we would say 5/2 is our radius of convergence. All right, well, I hope that clarifies the question there. Thank you so much for watching.

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