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

$ \frac {1}{3} + \frac {2}{9} + \frac {1}{27} + \frac {2}{81} + \frac {1}{243} + \frac {2}{729} + \cdot \cdot \cdot $

The series converges to $\frac{5}{8}$

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Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

let's determine whether the Siri's convergence of diversions and if it's conversion, will go and find the sun. So let's go ahead and rewrite this, my first just combining it's firms that have one in the numerator. All right, so that's the blue terms, all with the one. And then now we'LL combine the remaining terms, which all will have to in the numerator. So we have to over nine to over eighty one to over seven twenty nine and so on. Now let's go ahead and rewrite this So these are actually geometric and an easier way to see that this is just rewrite this. So we have won over three very cute three to the fifth and so on. So each time are multiplying by our equals one over three square and then for the green. We can go ahead and pull out to here, and then we have one over nine one over Nice where nine cute and so on. If you want, you could write. It is three square. It's just not necessary. So nine you could go ahead and do this and replace all the nines, but it's it's not required. So what's important here is that for the second series that we see that each time or multiplying by one overnight Hey, or if you want to write it in terms of three won over three square. So both of these ours are positive. They satisfy this one over nine in one over nine or both. One overnight is lesson one, so both of these will converge, and when you add two convergence series, the entire sum will converge. Therefore, the Siri's will be conversion. No. So if you add two convergence Siri's, the result was commercial. So some of two convergent Siri's is conversion. So all we did was rewrite this, um, and and Blue and Green and these air too conversion geometric series so the entire sum will converge, and then now we could do the second part. Let's change color here. Let's go to Red, find the song So we know for a geometric series. The sum is always the first term of the entire Siri's over one minus. Your common ratio are so for the blue. Siri's the first term we see that's one over three and then one minus R. And here are was just one overnight, so that's the first he reason for the second the first term. Well, first, let's pull out that, too. And then we see the first term is one over nine from this term over here, and then our are still one overnight. So let's just go ahead and simplify so we can write. This is three over mine, and then we add two over nine over here. That's gaming us five overnight. And the reason I'm combining those numerator XYZ because they have the same denominator it overnight and God and cancel those nights. We get five over eight. So this is the sum of our convergence, Siri's, and that's our final answer.