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JH
Numerade Educator

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Problem 36 Easy Difficulty

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{1 + (\frac {2}{3})^n} $

Answer

Diverges

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Video Transcript

let's determine whether the syriza's conversion or diversion and if it's conversion, will go ahead and find that some as well. So looking at this Siri's, this's not geometric. There's just no way for us to go ahead and rewrite. This is something like eight times Art of the Empower. So otherwise, if it was possible, would probably take a lot of effort and not be worth the effort. So one of the easiest has to try first when you're unsure of what to do is what the author calls test for diversions. And this test basically just requires you to look at the limit as n goes to infinity of this term here and that you're in the summation. So this is our limits of compute and in the limit two or three and will go to zero. So this lemon is just one. And if you're not convinced of this fact, here is and goes to infinity to over three. So in this case you're just multiplying by above point six six each time. And if you do that and you and go to infinity, you're going to get a number that's getting closer and closer to zero So that's why I just dropped this term. That goes to zero, and I just have one over one plus zero, and that's my one. However, if we like to use the test for divergence, we've just shown that the limit exist and the limit is not equal to zero. Therefore by this test, So let me take a step back here by the test for diversions. The Siri's is divergent. So the two conditions let me summarize that. So the answer's diversion. We use the test for divergence. We showed the limit exists. We got this number over here one, but that was not equal to zero. And that's how we conclude that the Siri's is that virgin. That's our final answer.