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JH
Numerade Educator

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Problem 18 Easy Difficulty

Determine whether the series converges or diverges.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {2}{\sqrt n + 2} $

Answer

Diverges

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Video Transcript

let's determine wether this Siri's converges or diversion. Here, let me call this a M, and then let me define Bien to be too over this word of that. So here are all used Limit comparison test. So for Lim comparison test, we look at the limit as n goes to infinity and over being. So let's go ahead and write that out. Here's R A. M so that that's who is not inside the radical and plus two. So that's an over bien and I should have written the limit in the front. So this is the limit is N goes to infinity. Cancel those twos and then here we can go ahead and divide top and bottom by the square event and will end up with one up top and then one plus two over radical and and then, as we take the limit to over radical and ghost zero, and then we're just left over with one now one. It satisfies this inequality. It's bigger than zero, but then it's less than infinity, and this is what's needed. If we want to use limit comparison test, you need a number that's strictly bigger than zero two zero not equal to zero. It's positive, and it cannot equal infinity. Otherwise, you can not use limit comparison test and you have find another method to use. So now t answer our question. We instead compare it with the easier questions. This is the whole point of the limit comparison test. We get to look at the some of the end instead. That's the sum over here. And here you can use the pee test from section eleven point three, and here you have P equals one half because of the square roots. So this will be diversion. And by Lim comparison, test our Siri's will also be diversion because the series that we compared it to the papers and that's our final answer.