5.6.2

[STARTS AT 4:15]

Instructor: Last example, use the root test for this when you have the series n equals 1 to infinity of 1 over n to the n. So, again, let's take Rho equal to the limit and goes to infinity of the nth root of 1 over n to the n. Now, if you notice, there's not actually an exponent on the numerator. However, I could easily write this, and that would now be true, doesn't change the value because 1 to any exponent is just 1. 

So, that would mean, and I should note I'm also dropping the absolute value, but I haven't written them for a few of these because the terms are all positive. OK, but for completeness I should probably put those in there. OK, go ahead and put that there. So, this would be equal to the limit of 1 over n, which is 0. 

Now, because that is zero, that means this. Converges absolutely. By the root test, series n equals 1 to infinity over n to the n, and again because converging absolutely means that it converges. Here we are. Now, this one actually should make a little bit of sense because as the exponent gets larger I mean it's acting like a p-series, but the exponent is going to infinity. So, as long as that's bigger than 1, which it always is, we have that.