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Problem 51 Hard Difficulty

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \arctan (\ln n) $

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Video Transcript

okay for this problem. Just remember what the graph of tan of ex looks like. All right, we have some vertical Assam coats happening here. And then there's going to be, you know, some stuff happening over here, off to the side. But the important part is what's happening between minus power to and power to Because rumor are tan is the inverse of tangent between minus power two and poverty. As long as we're between minus poverty and poverty to we're going to pass the horizontal line test, which means that we'll have an inverse. Okay, so are tan is the inverse two ten x and minus power, too to pie over too, right? And we don't include minus power to or power over to because because, you know, it's not really allowed toe plug in infinity, right? Can of X is not really defined that infinity at minus infinity. But certainly we can still talk about limits, right? So if we have the limit as tax goes to infinity of Ark tan of Axe, that that is still something that we can discuss, right? So if we think about what happens as X goes to infinity well, just remember that with inverse functions the X and A Y coordinates, you have to switch, right? So think about what happens with pan of X with pan of X. If we have that, the Y value goes to infinity. Hey writes The wide value goes to infinity. Then what happens to the X value? X value is goingto be approaching high over too. And with the inverse function, it's going to be the opposite. So if we have the X value going to infinity, then that means that the output is going to be power for two. Okay, so this is probably too something thatyou wantto remember instead of, you know, drawing this graph and trying to think about it every time. So you should, you know, have have this, remember, And as long as you know that, then this problems actually not that bad limit as n goes to infinity of Arkan of Natural Log of of In I think that's all we're working with here. Yes, okay. An ark ten. That's a continuous function. So again, we use this trick where we said it were allowed toe pull the limit inside of the function and then we end up with Ark Tan of infinity can. That's something that we said that we could just We could just refer to his pie over too. Okay, So for convergence, we want the limit to exist, and it should be some finite now you So poverty to is definitely some finite value. So the sequence converges, converges to pie over too.