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Numerade Educator

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Problem 36 Medium Difficulty

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

Answer

diverges

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Video Transcript

once again we want to look at limit as n goes to infinity of AM case we haven't in happening in the denominator, we all said a square root of an happening in the denominator. When will do the trick where we divide the top in the bottom by something we wanted by the top on the bottom by determine the denominator that's going to infinity the fastest. So in that case, we're going to be dividing the top and the bottom by ten. Then we'LL just have minus one of the n plus one of top and one plus square root and divided by end. So that's one over square root of n. So this is limit as n goes to infinity of minus one to the end class one divided by limit as n goes to infinity of one plus one over square devyn and again Well, I wouldn't limit on top over limit on bottom as long as we don't get, you know, infinity over infinity or something divided by zero. So it's not happening here. Something else bad does seem to be happening here. This bottom limit just goes toe one, and we're just going to be stuck with Lim as n goes to infinity of minus one to the end plus one. And this limit is going to be trouble because it's just gonna be switching between positive one and negative one, so the limit does not exist and therefore are sequences said to diverge.