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JH
Numerade Educator

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Problem 28 Easy Difficulty

Use the Root Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \left( \frac { - 2n}{n + 1} \right)^{5n} $

Answer

Divergent

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Video Transcript

let's use the root test to determine whether the series converges or averages. So here, let me go ahead and circle this term and call this my a n. The root test requires that we look at the limit and through of a n so recall the following fact. We can always write a radical and through as X to the one over and power. So here's our a n and then we're raising that whole thing to the one over and power mhm. And so we'll use another fact here about exponents. So here in the parentheses, we have and express and exponents a negative two and over n plus one, and that's all being raised to an exponent. So that's our baby. And then that is being raised to another exponent, one over end. So the rule is is you can just multiply those exponents. So, in our case, well, just multiply the one over N in the five end and the ends cancel. And here as well, we should technically be looking at the absolute value of a M. And you can see why this is important here because which will need the absolute value here. So let me go out and just put my absolute values back here. And then Now, well, you have this fifth power to deal with. So if we wanted, we could drop the absolute value. Now, just make sure that you dropped the negative as well. And then two to the fifth. It's 32 and we'll have an unfit there. But we also have an N plus one to the fifth, and you can you can rewrite these ends as an over and plus one to the fifth. And when we take the limit, that just goes to one to the fifth, which is equal to one. So in the limit does go to one, and we just have 32. This number is bigger than one. So by the root test, we conclude that the series diverges, So diversion is our answer.