When do sequences diverge

And I encourage you to pause this video and try this on your own before I'm about to explain it. So let's look at this first sequence right over here. So the numerator n plus 8 times n plus 1, the denominator n times n minus So one way to think about what's happening as n gets larger and larger is look at the degree of the numerator and the degree of the denominator. And we care about the degree because we want to see, look, is the numerator growing faster than the denominator?

In which case this thing is going to go to infinity and this thing's going to diverge. Or is maybe the denominator growing faster, in which case this might converge to 0? Or maybe they're growing at the same level, and maybe it'll converge to a different number. So let's multiply out the numerator and the denominator and figure that out. So n times n is n squared. And then 8 times 1 is 8. So the numerator is n squared plus 9n plus 8.

The denominator is n squared minus 10n. And one way to think about it is n gets really, really, really, really, really large, what dominates in the numerator-- this term is going to represent most of the value.

And this term is going to represent most of the value, as well. These other terms aren't going to grow. Obviously, this 8 doesn't grow at all. But the n terms aren't going to grow anywhere near as fast as the n squared terms, especially for large n's. However there are several ways a sequence might diverge. This clearly diverges by getting larger and larger This is clearly a divergent sequence but it may not be clear how to prove this formally.

To show divergence we must show that the sequence satisfies the negation of the definition of convergence. Therefore the sequence diverges. This seems to have been rather more work than we should have to do for such a simple problem. Learn More. Determining convergence and then finding the limit Example Say whether or not the sequence converges and find the limit of the sequence if it does converge.

Since we have a constant in the numerator and an infinity large value in the denominator, we know that??? We can conclude that the sequence??? Get access to the complete Calculus 2 course. Get started. Learn math Krista King August 18, math , learn online , online course , online math , calculus 2 , calculus ii , calc 2 , calc ii , sequences , series , sequences and series , convergence of a sequence , convergence , divergence , infinite limit.

Learn math Krista King August 18, math, learn online, online course, online math, geometry, trapezoids, midsegments, midsegments of trapezoids. We will need to be careful with this one. We will also need to be careful with this sequence. Also, we want to be very careful to not rely too much on intuition with these problems. We will need to use Theorem 2 on this problem. Therefore, since the limit of the sequence terms with absolute value bars on them goes to zero we know by Theorem 2 that,.

So, by Theorem 3 this sequence diverges. We now need to give a warning about misusing Theorem 2. Theorem 2 only works if the limit is zero. If the limit of the absolute value of the sequence terms is not zero then the theorem will not hold.

The last part of the previous example is a good example of this and in fact this warning is the whole reason that part is there. Notice that. So, be careful using this Theorem 2. You must always remember that it only works if the limit is zero. Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i.