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Numerade Educator

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Problem 20 Medium Difficulty

Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
$ a_n = 2 + \frac {(-1)^n}{n} $

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2

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Video Transcript

the first ten terms that we would have would be one five over two five over three nine over four nine over five we'LL be more helpful for us to have this in a decimal representation. So we'd have won two point five. Come on, one point six six six seven two point two five one point a two point one six six seven one point a five seven one at one point a a nine and two point one And now we want to plot these points. Okay, So for in equals one, we have the value of one and it looks like our values get upto two point five and that's the biggest that they'LL be so weaken Think about that when we're figuring out where to put our our tick marks here So we'll do it in increments of point five Okay for in equals one we have one So one one And for an equals two we have two point five in equals Through have one point six six six seven Hey! Any calls for We have two point two five and equals five one point eight in equal six, two point one six six six seven. So a little bit over two for in equals seven one point eight five seven one. Okay. And then we'LL stop there because I think we can probably see that pattern. Now, this looks like we have some waves here, but the waves are getting smaller and it looks like we're startinto flat now and approach some some particular number here. Okay, so the number that we should be approaching you should be too, right. So you can see that from these numbers here, one point eight eight nine. That is the number below two. But then the next number we have something that's a little bit above two. And as we go out farther, we'Ll still have numbers that are below two numbers that are above two. But they'LL be getting closer and closer to two. Okay, Looks like we do have Ah, Looks like we do converge, right? If it looks like you have some horizontal ask himto happening here, then that looks like you converge. Looks like you're going to converge toe whatever that horizontal Assam tote is Come here. Okay. And limit as n goes to infinity of a n. So that's two plus minus one to the end over end. That is definitely going to be too, because as n goes to infinity one divided by end goes to zero minus one divided by and is also going to go to zero. So doesn't matter this minus one to the end that we have here Doesn't matter if minus one to the end is acting like a positive one or a negative one If we're divided by ten and letting and go to infinity then that'LL go away and we'Ll just get too for that limit. Okay, so that agrees with what seems to be happening here in our graph.