Mixed Convergence Tests

[STARTS AT 13:33]

Instructor: Last question we have here is a mixed bag, to see what's going to happen with these. The sum n equals 1 to infinity of 2 to the n over 3 to the n plus n. OK, this is not familiar. This is not alternating. It might be similar to something. So let's see, term by term, 2 to the n over 3 to the n plus n. 

If this is less than or equal to a geometric, which in this case, it looks like it might be-- to 2/3 of the n. Let me write that differently-- the denominator in on the left is bigger, always, for all n. Therefore, that is going to happen. Now, this converges. In fact, it converges to-- let's see, our first term, that would be 1 over 1 minus 2/3-- so this actually converges to 3. We don't care what it converges to, because it doesn't help. 

But because this is less and the sum of n equals 1 to infinity of 2/3 to the n converges, we can say, therefore-- the sum from n equals 1 to infinity of 2 the n over 3 to the n plus n-- it converges, by the direct comparison test. And actually, on the last, that last one right there, it converges by the root test. 

So recognize when you need to use each of these tests, since we do have a whole bunch of them. Now, this next thing is a summary. And I'm just going to scroll over this pretty quietly, going to scroll through this because this is useful to have. It's a table that's in the book. I would suggest you actually go to the book and look at it. Go to this URL. 

But the divergence test, geometric series, those are two that are familiar, geometric and p-series. Next, we talked about the comparison and limit comparison test. With the divergence test, we discussed the integral test. And I, again, suggest you go back and read these and make sure you're comfortable with using whichever one. Integral test, then we have our alternating series test, which only works for alternating series. Then we have the ratio and the root test, the last two that we discussed. 

That is the end of this section. Hopefully, you can now feel a little bit more familiar with all of those different tests. That's it.