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Numerade Educator



Problem 55 Hard Difficulty

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ a_n = \frac {n!}{2^n} $




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Video Transcript

for this problem. Recall that in factorial is just one times two times three times. Stop that thought all the way up to times in and to the end is just two times two times two times two where there's in total copies of two there. So if we rewrite this, it's one half times two over too. Times three over, too. Not got times in over two. So if we get some bounds on this, we know that this is going to be greater than one half times to over two times three over to and all of the terms over here, all those terms, they're going to be bigger than three over to. So if we just replace all of them with three over to, then we're going to get something smaller, okay? And there's a total of in minus two terms over there. Right, Because we have it. We started with in terms total, and then we have this one half term, and we have this to over two term. So after taking those out, we have in minus two terms left. Okay, so we get this bound happening here, and what that means is that if n goes to infinity. That's going to be greater than limit as N goes to infinity of one half times, two over to two or two is just one times three over, too, to the end, minus two three over two is bigger than one so through over to is bigger than one. And that tells us that this term, through over to to the in minus two power is going to blow up to infinity. So certainly multiplying by one half, it's still going to be infinity. So if our AI in terms has in goes to infinity, are bigger than infinity than they have to blow up a swell right. So we also have to have that limit. As n goes to infinity of Anne, that also has blow up in his infinity, so our sequence diverges.