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JH
Numerade Educator

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Problem 17 Easy Difficulty

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
$ 3 - 4 + \frac {16}{3} - {64}{9} + \cdot \cdot \cdot $

Answer

DIVERGENT

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Video Transcript

let's determine whether this to geometric Syriza's conversion or not. And then, if it's conversion, will go ahead and find the sum. So we know that a geometric Siri's, which is usually written in this form, doesn't have to start at one, but it usually does. Then this will converge on ly if that flu value bar is strictly less than one and it'LL divers, otherwise so absolute value are bigger than or equal to one. So let's find what the R is here. And once we find our will, just rely on this fact here to give us our answer. And if it happens to emerge, well, go ahead and use the formula for the geometric series to find the sun sonar problem. How do we find our well in general? Here's howto find R. If you just go ahead and take any two consecutive terms in the sequence of plugging some end than N plus one and then if you go ahead and divide the second term over the first term over the previous term, everything will cancel out except our so if you ever want to find our and the sum is not given, signal notation just take any element you want except the first one, Let's say, for example, minus four and then just divided by the one right before in our case, dashes three you didn't have to use negative for you could have used sixteen over three, and then you would divide that by negative four. In either case, you will still get minus for over three. So this is our the It's also called the Common Ratio, and you could see why it's called common ratio because dis fact over here. So by this fact, over here we see that the absolute value of our absolute value of negative for over three, that's for over three. That's bigger than one, so it will diverge. So I guess, in summary, our answer is diversions.