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JH
Numerade Educator

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Problem 17 Medium Difficulty

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

Answer

Converges

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Video Transcript

I must test the series for conversions or divergence. Now the first thing I noticed is that the syriza's alternating This's just due to this minus one to the end, hanging out the front. So this suggests we can try to use the alternating serious test. So there we can take bien Teo. Just be the basically the absolute value, this part over here. So it's just absolute value. Sign pi over end. And actually, we can argue that this will. You could drop the absolute value here because first of all, this pyre N is always bigger than zero because positive over positive is bigger than zero and the biggest it could ever be when an equals one is pie. Therefore, if we think of this is like in the unit circle, this is basically saying that pie a baron is. It could be over here above the origin or could go all the way to pie. So this is our our angle pie of Ren and the Unit Circle, therefore were in either quadrants one or two, and therefore we know that sign of pie over end will be bigger than zero. And that's important here because that's one of the conditions for BM in the alternative theory says there has to be positive numbers in this case is greater than or equal to zero you can. Being equal to zero is not a problem. You just don't want negatives. Now. The second condition here is that the beings are going to zero in the limit. And this is true because as n goes to infinity, we could just use the fact that signs continuous to just push the limit on the inside. And that's just becomes tire infinity, which is zero. So sign of zero equals zero, and now there's one more condition to check. That's if BN is decreasing. So this question is whether sign So we wanted to check it and plus one is less than or equal to B N. That's what it means to be decreasing Sobhi and plus one a sign of pie over N plus one. And the question is, is this less than or equal to sign a pile? Bryn? And here we could just explain that this is true using some geometry. So if this is pie around over here, when you look at pi over and plus one This is a smaller angle because as a bigger denominator, so eventually when we go into the limit is and his large, however, and will be somewhere over here. But then, when we increase and buy one, we're getting an even smaller angle. So the reason I'm drawing this in the first quadrant is eventually and will be bigger than pie and is going to go to infinity, so it's eventually bigger than pie. It's bigger than four, so pyre over end will be between zero and pi or two. This is why I'm drawing it in quadrant one. This is it, and it's bigger than by. So this is Yeah, this is why we're drawing an enclosure one. And as we saw as we went from blue to red. So here's blue and then here's red. Will we increase and buy one? The angle gets smaller, and so that means that the sign is getting smaller. And this is why this inequality is true. So all three conditions one, two and three for the alternating serious tests have been checked. Therefore, the Siri's converges by the ocean aiming Siri's test, and that's our final answer