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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. So for very, very large n's, this is really going to be approaching n squared over n squared, or 1.
So it's reasonable to say that this converges. So this one converges. And once again, I'm not vigorously proving it here. Or I should say I'm not rigorously proving it over here. We then say that zero is the limit or sometimes the limiting value of the sequence and write,.
This notation should look familiar to you. It is the same notation we used when we talked about the limit of a function. Using the ideas that we developed for limits of functions we can write down the following working definition for limits of sequences. The working definitions of the various sequence limits are nice in that they help us to visualize what the limit actually is. Just like with limits of functions however, there is also a precise definition for each of these limits.
Now that we have the definitions of the limit of sequences out of the way we have a bit of terminology that we need to look at. So just how do we find the limits of sequences? Most limits of most sequences can be found using one of the following theorems. This theorem is basically telling us that we take the limits of sequences much like we take the limit of functions.
We will more often just treat the limit as if it were a limit of a function and take the limit as we always did back in Calculus I when we were taking the limits of functions. So, now that we know that taking the limit of a sequence is nearly identical to taking the limit of a function we also know that all the properties from the limits of functions will also hold.
These properties can be proved using Theorem 1 above and the function limit properties we saw in Calculus I or we can prove them directly using the precise definition of a limit using nearly identical proofs of the function limit properties. Next, just as we had a Squeeze Theorem for function limits we also have one for sequences and it is pretty much identical to the function limit version.
This will be especially true for sequences that alternate in signs. If we say that a sequence converges, it means that the limit of the sequence exists as??? If the limit of the sequence as??? A sequence always either converges or diverges, there is no other option. I create online courses to help you rock your math class.
Read more. There are many ways to test a sequence to see whether or not it converges. Sometimes all we have to do is evaluate the limit of the sequence at??? If the limit exists then the sequence converges, and the answer we found is the value of the limit.
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