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JH
Numerade Educator

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Problem 18 Easy Difficulty

Test the series for convergence or divergence.
$ \displaystyle \sum_{n = 1}^{\infty} (-1)^n \cos \left( \frac{\pi}{n} \right) $

Answer

diverges

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Video Transcript

Let's test the Siri's for conversions or diversions. So first, let's look at the limit is N goes to infinity of co sign off Pi over end. A Zen gets really, really large. The fraction will get closer to zero. So we just take the limit. You get coast on zero, and that's one. So this tells us that Well, what's what will happen when we start multiplying by minus one to the end? This just means that the limit is n goes to infinity of an was called this entire thing here this entire term and negative one to the end. Co sign pi a grin. Well, a Zen gets really, really big. We know the co signs getting closer, the one so that just means that this will just continue to oscillate between negative one and one so the limit does not exist. So by the diversions test, and then the reason reusing the test is we've just showed the limit is N goes to infinity. Negative one to the end. Co sign Pi over in is not equal to zero. That's why we're using the diversions tests. Because of the non zero, we can conclude that our Siri's diverges and that's our final answer