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Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {6 \cdot 2^{2n - 1}}{3^n} $

The geometric series diverges

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Missouri State University

Oregon State University

Baylor University

Let's determine whether this geometric Siri's converges or diverges and if it's come urgent will go ahead and find the summer as well. So geometric series is usually ran in the form you have, like some number A here, and then you have some term R Let's constantly being well supplied. And so the Siri's could start at any number of many times. Like in our problem, it's one and it goes up to infinity. So let's just rewrite our problems so that it looks like this. So I have six here and let me rewrite this. We have two of the two groups two and minus one. So this is to the two and over to to the one which is to square to the end over, too. So let's write that here we have two square and over to, and then we're still dividing by three end. Let's simplify six over to that's history and then combining these two terms right here we have for over three to the end. So now this were in this form here, and we see that are equals four or three a equals three. However, we know that a geometric series so geometric. Siri's converges on ly if the absolute value of our is less than one. But in our case, since the absolute value are equals for over three is bigger than one, the Siri's will diverge, so the Siri's is diversion, and that's your final answer.