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JH
Numerade Educator

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Problem 30 Easy Difficulty

Use the Root Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 0}^{\infty} $ (arctan $ n)^n $

Answer

Divergent

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Video Transcript

let's use the root test to determine whether the Siri's conversions air averages the roots has requires that we look at the limit and goes to infinity and through absolute value of an We're in our case, a N is given by this formula over here. So go ahead and use the following facts from algebra Regarding the end through, we could always rewrite this's X to the one over and Sol, go ahead and do that over here, not formula. So have ark tend of end to the end power and also we're dealing with absolute valley here. So let me just put the absolute value inside. And this is all being raised to the one of rent. And so we're dealing with the exponents in the following form. So we'LL use another fact about exponents when you raise an exponents. Ooh, another. You just multiply those two exponents. So here, when we multiply, be in the sea we just get one and then we just have the limit and approaches infinity absolute value arc tan. And if you could recall from the graph here when you take the limit of the Ark tan, it approaches one of its aspirin toads. So here's a rough sketch of the Ark tan graph. Lucy. So you can see from the graph of Arc Tan that is, the limit goes to infinity. The Y value approaches pi opportune. However, this number is larger than one, and so by the root tests we conclude that the Siri's diversions