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Numerade Educator

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Problem 48 Medium Difficulty

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \sqrt[n]{n} $

Answer

The sequence converges to 1

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Video Transcript

you have a problem that could be a bit tricky the first time you see it. But it's another problem where you just want to rewrite a You're a a term. So instead of thinking about it as into the one over in, you can think about it as e to the natural log of end to the one over end, right into the natural log event that's going to be the same thing is just in. And this is the same thing as natural log are starting something I need to the natural log of n divided by n so as n goes to infinity a n is Lim has been goes to infinity of E to the natural log of n divided by n exponential function is continuous. We can just pull the limit into the exponents. And now this limit that we have a Pierre. This is, ah problem for low Patel. Everyone just plug in and equals infinity. Then we'd have infinity over infinity, which is certainly not allowed. But if we are in that set up and we're allowed to use live Patel's rule Patels lure, it means we do the derivative of that new neighbor divided by the derivative of the denominator. So the drug of the numerator you get one over in Not Traugott in has derivative one over in and the derivative of end with respect to end is just one says in goes to in family. Our exponents is now going to go to zero. So we have zero, which is one. So we do converge, we converge to one.