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Numerade Educator

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Problem 40 Medium Difficulty

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac {\tan^{-1}n}{n} $

Answer

The given sequence converges to 0

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Video Transcript

first recall that the candidate function has a graph that looks like this. Oh, okay. The important part here is this segment here between minus pi over two and pie over to so notice that as we go up to infinity over here that the X value gets close to pie over too. Remember our can Our ten is the inverse of Qianjin between minus power two and power to Okay, So for the inverse functions the X and A Y coordinates air switched So here we go up to infinity X goes to poverty to so for our ten if we're looking at Ark Tan of infinity in the output would be pi over too can The important thing here is that this is just some finite number So when we're taking the limit as in goes to infinity of a end, that's limit as in goes to infinity of Ark tan we just use the same notation So this is just another notation for our tent art can of n divided by n We knew the trick where we dio limit on top over limit on bottom So we're allowed to do that as long as we don't get infinity over infinity or something divided by zero. So here, women on top, as we mentioned, that's going to be poverty Tuesday. Just some finite number limit on the bottom is going to be infinity. So this is some finite number divided by infinity, and that limit is zero. So we do converge. We converged to zero.