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