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

$ 2 + 0.5 + 0.125 + 0.03125 + \cdot \cdot \cdot $

The geometric series converges and the sum is $\frac{8}{3}$

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Campbell University

University of Michigan - Ann Arbor

Boston College

determine whether that she had metric. Siri's is converting our diversion, and if it's conversion, find this home. That won't be a problem because we do have a formula for geometric Siri's and here they're giving you the Siri's. So before we look at the serious and detail, let's just go to the side and recall the definition for geometric. This means that you have Ah, some of the form A are to the end. Some people might right they are and minus one. It doesn't matter here whether and also I'm not writing the starting points here, it could be one. It could be zero it actually, it could be any number of I won't even write anything down there. Basically, the idea is each time and increases by one. You're multiplying by the same number each time, and it's always are r. So in this problem they're telling what we know that this is geometric. What we have to find out what our is and what is our A. So here to find the r noticed that if you take this if you take one term and you divided by the previous term, your leftover with are so If you ever want to find our and from a geometric series, just take some term that you see anyone that you like and divided by the one right before. For example, Here I look at the seconds or I'm divided by the one right before it, so I get a half over to which is the fourth. So that's my r and you could check. You could use this for any two values you could use instead of using to one point five. You could use point five and point one two five. If you divide point one two five over playing five, you'LL still get a fourth, so that is our are. And then here we can go ahead and use a to just be to that first term. So here, since our satisfies the following inequality, it's less than one. The Syrians will converge, So this is the condition that you need for convergence. Otherwise diverges even if it's equal to one diverges. Now let's use our formula for the geometric series. So here we have a over one minus R. Or I think the easiest way to memorize this is the first term. If you use this formula here it. This won't depend on what you're starting Point is, And in our problem, that's just two over one minus a fourth. So we have to over three over four, and then you could go ahead and simplify that Teo eight over three. And that's your final answer for this geometric series.