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Numerade Educator

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Problem 37 Medium Difficulty

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ \left \{ \frac {(2n - 1)!}{(2n + 1)!}\right \}$

Answer

Converges

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Video Transcript

we're the same thing that we're doing before. It's just written a little bit differently now to the A end. Terms that were looking at her two in minus one fact will divided by two in plus one fact oil and wondered the same thing we were doing before. We want to look at the limit as in goes to infinity of a n well, whether or not that women exists. And if that women exists and is finite than our sequences, said that that is said to converge. Excuse me. Otherwise, that is said to diverge. So before we take the limit, weaken rewrite this expression a little bit. So this is two in minus one. Fact will and then another way that we can write too. In plus one, factorial is two in minus one factorial times two in times to end plus one. And this is a convenient way to write it because now we have ah, nice cancellation. That happens right here. And we see that this simplifies to one over to in times two and plus one. And now when we do limit as n goes to infinity of a n r. Expression becomes easy to work with. We just have one over to end times two and plus one. So certainly the denominator is going to go to infinity. The numerator is just one. So this is one over infinity, which is zero. So our sequence does converge, converges to zero.