Comparison tests for convergence
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Worked example: limit comparison test
- [Instructor] So we're given a series here, and they say, "What series should we use "in the limit comparison test?" Let me underline that, "The limit comparison test, "in order to determine whether S converges?" So let's just remind ourselves about the limit comparison test. If we say, if we say that we have two series, and I'll just use this notation, a sub-n, and then another series, b sub-n. And we know that a sub-n and b sub-n are greater than or equal to zero for all n, for all n, if we know this, then if, then if the limit as n approaches infinity of a sub-n over b sub-n is equal to some positive constant. So zero is less than that constant is less than infinity, then either both converge, then both converge, both converge, both converge, or both diverge, both diverge. And it really makes a lot of sense, because it's saying, look, as we get into our really large values of n, as we go really far out there in terms of the terms, if our behavior starts to look the same, then it makes sense that both these series would converge or diverge, and we have an introductory vizio, (chuckles) we have an introductory video on this in another video. So let's think about what, if we say that this is our a sub-n, if we say that this is a sub-n right over here, what is a series that we can really compare to? That seems to have the same behavior as n gets really large? Well this one just seems to gets unbounded. This one doesn't look that similar. Has a three to the n minus one in the denominator, but the numerator doesn't behave the same. This one over here is interesting because we could write this, the is the same as a sum n equals one to infinity. We could write this is two to the n over three to the n, and these are very similar. The only difference between this and this is that in the denominator here, or in the denominator up here we have a minus one, and down here we don't have that minus one, and so it makes sense given that that's just a constant, that as n gets very large, that these might behave the same. So let's try it out. Let's find the limit, and we also know that the a sub-n's and the b sub-n's, if we say that this over here is b sub-n, we say that's b sub-n, that this is going to be positive, or this is going to be greater than or equal to zero for n equals one, two, three, so for any values this is going to be greater than or equal to zero, and the same thing right over here. It's gonna be greater than or equal to zero for all of the n's that we care about. So we meet these first constraints, and so let's find the limit as n approaches infinity of a sub-n, which is, I'll write in that red color, which is two to the nth power over three to the n minus one over b sub-n, over two to the nth over three to the nth. So let me actually do a little algebraic manipulation right over here. This is going to to be the same thing as two to the nth over three to the n minus one, times three to the n, over two the n. Divide the numerator and the denominators by two to the n, those cancel out, and so this will give us, this will give us three to the n over three to the n minus one. We can divide the numerator and the denominator by three to the n, and that'll give us one over one minus one over three to the n. So we could say this is the same thing as the limit as n approaches infinity of one over one minus one, over three to the n. Well what's this going to be equal to? Well as that approaches infinity, this thing, one over three to the n, that's just gonna go to zero. So this is, this whole thing is just going to approach one. And one is clearly between zero and infinity, so the destinies of these two series are tied. They either both converge or they both diverge. So this is a good one to use a limit comparison test with, and so let's think about it. Do they either both converge, or do they both diverge? Well this is a geometric series, our common ratio here is less than one, so this is going to converge, this is going to converge. And because this one converges, by the limit comparison test, our original series S converges. Converges, and we are done.
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