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Numerade Educator

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Problem 53 Hard Difficulty

Determine whether the sequence converges or diverges. If it converges, find the limit.
$ \left \{ 0, 1, 0, 0, 1, 0, 0, 0, 1, . . . \right \} $

Answer

diverges

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Video Transcript

for this problem, the possible outcomes or that we have divergence that we converge, in which case we would converge to zero or we would converge toe one. We wantto look at the possible cases for a convergence. So if Anne was equal to one, then a and plus one zero by that would just mean that there is no consecutive ones that show up here. Therefore, limit as n goes to infinity of a N is not going to be equal toe one tense. If this limit existed, we should have. We should have that. It's the same thing is if we just have the n plus one. Okay, so this is That's the idea. If Ltd's in, goes to infinity of and exists, then it should be the same thing as Ltd's in Goes to Infinity of A and Plus One. Okay, so we use the same type of logic toe rule out the possibility that es and is equal to zero. Right now, we've established that the limit is not one. Okay, so that's the easier part. And now we noticed that for everything in reverie end such that a vin a zero there exists a positive number M and we'LL write in sub end because this positive number is going to depend on what Innes So they exist in him sub in such that have in plus him seven is one. Okay, so now we can use the same type of logic that we use before. So we know that limit as n goes to infinity of a N is not going to be equal to zero since if it wass, we'd have we have that limit as n goes to infinity of a in plus M and is one case that's just by definition, by definition of m ine. Ine is defined toe have this property. Yeah, but as n goes to infinity, we certainly have that in plus m seven goes to infinity because remember himself in is just some positive number. Therefore, if limit as n goes to infinity of a n existed, we'd have same thing we had before. That limit, as in goes to infinity of a end should be the same thing as limit as n goes to infinity of a in plus and seven Okay, But as we discussed, if the limit as n goes to infinity of a and zero, then, by definition, this limit is going to give us one case that contradicts what we would want to happen here. So since we do not have inequality, that would happen. We know that the limit can't possibly the zero. And we've already mentioned that the limit can't possibly be one. So therefore, the only other option is that sequence diverges.