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Problem 31 Medium Difficulty

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac {4^n}{1 + 9^n} $

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Video Transcript

So we look at limiters and goes to infinity of am. If this limit exists and is finite, then the sequence is said to converge. Otherwise, that is said to diverge. So again, we used a trick where we look at the denominator and we look at the term that's growing to infinity, the fastest. And then we divide the top and the bottom by that. So in this case, we're going to divide the top in the bottom by nine to the end for the end divided by nine to the end. And then down here we have one over nine to the end, plus one. And then we could just rewrite this a little bit. This is Lim is n goes to infinity of four, divided by nine to the end, divided by one over nine to the end, plus one. And then you, Khun, just put the limit on top and the limit on bottom. As long as you don't get something an indeterminate form like infinity over India or something divided by zero, then you could just put the limit on top and then a limit on the bottom. So the top limit, as in goes to infinity for over nine to the end. That's going to go to zero, since for over nine is less than one in absolute value. And then in the denominator this term here, that term is going to go to zero because nine to the end is going to go to infinity. So we'LL have won over infinity and then we're just gonna be stuck with the one. But that's fine. Zero over one is not indeterminate form. So no issues here, So we just get zero for our limit. So it does converge, converges to zero. Okay, again, we use the fact that for over nine is less than one in absolute value. That was what allowed us to conclude that this term was going to go to zero.