And diverge means that it's not approaching some value. So let's look at this. 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.
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.
If we multiply it times negative 1 to the n, then this one would be negative and this would be positive. But we don't want it that way. We want the first term to be positive. So we say negative 1 to the n plus 1 power. And you can verify this works.
When n is equal to 1, you have 1 times negative 1 squared, which is just 1, and it'll work for all the rest. So we could write this as equaling negative 1 to the n plus 1 power over n. And so asking what the limit of a sub n as n approaches infinity is equivalent to asking what is the limit of negative 1 to the n plus 1 power over n as n approaches infinity is going to be equal to?
Remember, a sub n, this is just a function of n. It's a function where we're limited right over here to positive integers as our domain. But this is still just a limit as something approaches infinity. And I haven't rigorously defined it yet, but you can conceptualize what's going on here. As n approaches infinity, the numerator is going to oscillate between positive and negative 1, but this denominator is just going to get bigger and bigger and bigger and bigger.
So we're going to get really, really, really, really small numbers. And so this thing right over here is going to approach 0. Now, I have not proved this to you yet. I'm just claiming that this is true.
But if this is true-- so let me write this down. If true, if the limit of a sub n as n approaches infinity is 0, then we can say that a sub n converges to 0. That's another way of saying this right over here. We do, however, always need to remind ourselves that we really do have a limit there! If the sequence of partial sums is a convergent sequence i.
Likewise, if the sequence of partial sums is a divergent sequence i. To determine if the series is convergent we first need to get our hands on a formula for the general term in the sequence of partial sums. So, to determine if the series is convergent we will first need to see if the sequence of partial sums,. The limit of the sequence terms is,. So, as we saw in this example we had to know a fairly obscure formula in order to determine the convergence of this series.
In general finding a formula for the general term in the sequence of partial sums is a very difficult process.
We will continue with a few more examples however, since this is technically how we determine convergence and the value of a series. This is actually one of the few series in which we are able to determine a formula for the general term in the sequence of partial fractions. Therefore, the series also diverges. Again, do not worry about knowing this formula. The sequence of partial sums is convergent and so the series will also be convergent.
The value of the series is,. As we already noted, do not get excited about determining the general formula for the sequence of partial sums. Two of the series converged and two diverged. Notice that for the two series that converged the series term itself was zero in the limit.
This will always be true for convergent series and leads to the following theorem. Then the partial sums are,. Be careful to not misuse this theorem!
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