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JH
Numerade Educator

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Problem 35 Easy Difficulty

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent.

$ \displaystyle \sum_{n = 2}^{\infty} \left( \frac {n}{\ln n} \right) ^n $

Answer

Absolutely divergent

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Video Transcript

let's use the root test to see whether the Siri's diverges. So if we call this term here and the root test requires that we look at the end through absolute value and so on our case, let's rewrite this Andrew as the one over and power. And then we'LL have and over natural log of end to the end power. And here there's no need to write the absolute value because and the natural log of end or both positive if Anne is bigger than or equal to two, which is known from our summation, and they're using your laws of exponents. If you raise exponents to another, exponents will multiply those exponents. So here that will give us and over natural log of end. And then in the roux tests, we want to take the limit of this, and if it helps, you can go ahead and replace and with X and use Lopez house rule. So here, go ahead and use the X here. So exes any rule number so we could take a derivative for us before and was just positive image or so the derivative is not defined, so we'd use low Patel's rule here you get one over one over X so Lim X goes to infinity of X. That's infinity, and that's bigger than one. So we conclude that the Siri's diverges by the root test.