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Numerade Educator



Problem 39 Hard Difficulty

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ \{ \sin n \} $




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Video Transcript

Okay, So if limit as n goes to infinity of sign of in exists so if it does exist, then for any Aunt Inger say, am we should have that Linda, as n goes to infinity of sign of in that should be the same thing as limit as n goes to infinity of sign of in plus n Because if in goes to infinity and am is just some inter jer than certainly in plus M is going to go to infinity as well. So if this limit truly exists, then these two are going to be the same thing. Now, if we think about exactly what sign of in is we can think about the unit circle. I remember one of the definitions that we have for sign of data is that if we're on the unit circle here, sign of theta is just going to be the y Coordinate for this point over here. Okay, so now think about any point that we pick on our circle. Say we pick some point over over here, Then if we ended up adding say for tow our angle, then we go all the way around and, you know, end up end up somewhere over here. So adding four could change your sign or if we're really close to here than if we added for that might push us over the edge and go back around here because remember, papayas three point one four that so if you were close enough to hear and we added before, we might go too far around and still be some positive number. But if we were too far over here and we added three, then we, you know, end up somewhere over here, and we'd still have a change in sign. So certainly if if you have a change in sign, then the two things that you're comparing cannot be the same. So either adding three or adding four is going to cause for there to be a sign change between sign of in and sign of in plus three or four. Okay, so what we mean is that we're going to have that either sign of end. It is not equal to sign of n plus three or depending on where exactly we are on the circle. We might have that sign of N is not equal to sign of and plus four, right? And this This should be clear just from, you know, thinking about some position on the unit circle and thinking, you know, certainly there's going to be some integer that I can add that's goingto cause me to change signs. And if there's any interview that you can add that is going to cause you to change signs, then there's no way that you can have an equality happening. Okay, so that that contradicts what we're saying here, right? If you just throw limit signs around everything, then you know this condition is going to fail for some into your M. Okay, so slap on these limits and this contradicts what we were saying before, right? We said that if the limit did exist and for any integer that we put on here, we should have inequality happening. But as we've seen for some integer, there is going to exist some inequality. So I'm sorry that there's not any real great explanation for this problem. It is definitely a problem that you have to think about, and it's an unusual argument, but that's the best that I can come up with at the moment and the the final conclusion is that we do not have convergence sequence diverges. And don't don't worry too much, if you know if this problem was a little bit over your head. This seems like a tricky problem that really requires a lot of thought and thinking about what Sine is, and definitely for me helps toe think about this unit circle and just, you know, think about sign as being the why co ordinate on the unit circle and imagining that, you know, Certainly there's going to be some integer that I can add that is going to cause me toe go around and change sign, and if you change sign, then you're certainly not going to be equal, and that's that's the point.