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Problem 59 Hard Difficulty

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .)
$ a_n = \arctan \left( \frac {n^2}{n^2 + 4} \right) $

Answer

converges

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Video Transcript

okay for this problem. If you were starting to grasp this, you plug in the first value of N and equals one. You get arc tan of one fifth. And the only way to deal with that is just by plugging it into your calculator you get, you know, you get approximately zero point one nine seven. Okay, so then you could could clot that you will make this point too. Point four point six point a. So when it is one where about over there? And then you, Khun, plug in and equals to put in your calculator. And you're going to get something that that looks Looks like this okay. Might be a little bit more steeped in that, but something that looks something like that. So if you were just doing this by hand, you just plug in different values of end quote in your calculator, make a bunch of different dots, and then connect the dots and start to see exactly what this thing looks like. So you want to be looking for horizontal Assam totes here. So it looks like we were getting some type of pores on last until happening, right? Looks like our graph is starting to flat now as we go further to the right, so it it does look like we converge. And if you're just taking a total guess here you would, you know, guess that it's somewhere around point eight. Converge somewhere around point eight, right? So it's hard to say if it's exactly point eight just from looking at the graph, it looks like it's gonna be somewhere around there. But if you want to figure out exactly what it wass, then you could you know, you do the trick where we we say that is long as our function is continuous, then we can pull the limit inside. So if we're looking at, Lim has n goes to infinity of Park ten in, squared over in squared plus four. Like I said, we do our trick where we pull the limit inside, okay, and then this. I think we should know how to handle by now. If you're really tryingto do every single step from here, you would divide the top in the bottom by in squared, and you would see that we're going to end up with Ark Tan of one. So our ten of one is pi over four, and that turns out to be pretty close toe. What were suspected it was gonna be his power for is about zero point seven eight five. So it's it's pretty close to zero point eight. So we we do converge and we converge to pi over four.