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

$ 4 + 3 + \frac {9}{4} + \frac {27}{16} + \cdot \cdot \cdot $

The series converges to 16

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let's determine whether this given geometric series is convergence or die Apology and if it's conversion, will go ahead and find the sun so we know that it's geometric that's given. And we could just look at the series to find this so so geometric Series is usually run in the form et arses and and can start at one, but it could also start it any number. So here we like to say what they are is because if we know what our isn't our problem for this problem over here, If we know what our is, then we know that if absolute value are less than one, then the theories is conversion. Otherwise, which means absolute value bar is bigger than or equal to one, then diversion. This is why we need our So let's go and find that are for our problem. So the way to find our is well for geometric series. Each time we go from one term to the next, which is multiplied by a factor of our So we have four equals. Three times are excuse me. Three equals four times are so it's geometric. So to get from for the first term to the second we take the first term we multiplied by r, and that gives us to the next. So go ahead and solve that for r R equals three over, for this is the R value that we want you. Now we're basically done. Absolute value are equals three over four, which is less than one. So all our Siri's the Given one up here, our Siri's converges further. We have more work to do here because of this additional sentence. If it's kam urgent, what is true? Let's go ahead and find the song. So recall that the sum from geometric series we have a formula for this some let me rewrite original, then we know for geometric. It's the first term of the series over one minus. R. You could always use this formula here for geometric. So the first term with to see that's for So that's a foreign, the numerator and then one minus R, which was three or four by our previous work over here. And let's just simplify this and we basically have our answer. We have one over four and the bottom that becomes sixteen in Flore. Let's summarize this by saying that they're given Siri's converges and the sum of the Siri's is sixteen, and that's our final answer