5.4.2

[STARTS AT 9:15]

Instructor: And lastly, we will compare-- use this one. 5 to the n over 3 to the n plus 2 using the limit comparison test. Now, this appears to match up with one of our geometric sequences. So bn, we'll choose 5/3 to the n. Now, that diverges. 

That diverges because 5/3 is greater than 1. So let's compare these two, again, term by term. So we have 5 to the n over 3 to the n plus 2 or 5 to the n over 3 to the n. All right. Let's see. That will be 5 to the n over 3 to the n plus 2 times 3 to the n over 5 to the n. 

And that limit once we rewrite that algebraically, that is 5/3 to the n over 3 to the n plus 2. That limit is 1. Now, if the limit is not 0 but it's a constant, then they both converge or both diverge. 

So by the limit comparison test, since the sum 1 to infinity of 5 over 3 to the n diverges, the sum from 1 to infinity of 5 to the n over 3 to the n plus 2 diverges as well.