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JH
Numerade Educator

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Problem 37 Medium Difficulty

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

$ \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1)^n \arctan n}{n^2} $

Answer

Absolutely Convergent.

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Video Transcript

Let's show that this series is absolutely convergence, so here will go ahead and look at a similar Siri's. But this time will take absolute value. So here there's absolute value. So all that will do for us is just get rid of the negative one to the end power. And let me let me explain why here? This is because and square in our town of n are both positive in here. So technically, Arc Tan could be negative. You think of the graph of this, but as long as and is positive than Arc Tan is positive, so that justifies Why weaken, drop the absolute values and then this Siri's. We know this is less than or equal to the largest that the Ark Tan could be. Is this horizontal Assam towed at pi over too. So just go ahead and replace Arc tan with pie over too. And here you can pull out, pull off their constant. So we're using a few test here. You could see from the inequality that were setting ourselves up to use the comparison test. So now we have to look at our upper bound this new series over here blue and that's a P Siri's. So the Siri's converges you would apply the Pee test with P equals two, and that is bigger than one. So the Siri's will converge. So now the upper bound converges. So by the comparison test, the lower bound also converges, and we conclude that the Siri's that the original Siri's is absolutely commercial. So let me just go ahead and call this instead of writing this whole thing down again. Let's just call this star. The Serie Star converges. Excuse me, is absolutely commercial.