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Numerade Educator

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Problem 6 Medium Difficulty

Use the Integral Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{\left(3n - 1 \right)^34} $

Answer

The given series converges

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Video Transcript

So to use the integral test, we will evaluate the integral and find out if it is conversion or divergent. So, in a girl from one to infinity of one over three X minus one to the third power time, An extra form the bottom and we will take a limit. Ah, and at the same time our well, Sorry. Yeah, I got it myself. Here. We can see that we're probably gonna do some kind of a u substitution. If you see how to do that, you're already That's fine. But all the substitution for the sake of completeness here. Ah, specifically with these substitution of new equals three X minus one. So do you equals three the ex. So we now have ourselves in limit. S t goes to infinity of since we changed our variable I should change my bounds of integration. So it's leaving something generic for now we have Ah you two the negative. Third times one fourth times Do you over three since again we have that three up here when we replaced our variable. So that will give us the limit. He goes to infinity of negative one half times one fourth times one third you to the negative too. From a to B, and replacing our ex is back in and multiplying that fraction in the front. We end up with this here and we can see from here that now we have our negative exponents. All of our variables are under a negative exponents, which tells us that is we take the limit to infinity. We should get something convergent. So that would be Lemonis. Dios Infinity of negative one over twenty four and a three team minus one. The negative too. Minus two. You have to And this piece here goes to zero. So convergence.