6.1.2

[STARTS AT 9:11]

Instructor: Find the interval and radius of convergence of the sum of the series x to the n over the square root of n. All right. So our ratio test, x to the n plus 1 over the square root of n plus 1 divided by x to the n over the square root of n. Absolute value of that. And that limit reduces down to, we have the square root of n over the square root of n plus 1. And that would be just an x. Absolute value. The limit there is the absolute value of x. That converges when that will be less than 1, which means our interval of convergence is negative 1 is less than x is less than 1. And we can say our radius of convergence is 1. 

Now we need to consider our end points. So if x equals negative 1, then that is negative 1 to the n over the square root of n, which by the alternating series test converges. And again, just notice that that is a decreasing sequence there. It's decreasing. And the sequence itself goes to 0, the 1 over the square root of n. So we are going to include negative 1. If x equals 1 that is 1 over the square root of n, which as a p series, that diverges. So we will not include one. 

So our interval of convergence, we can write that as negative 1 to 1, not including 1, including negative 1. 

[ENDS AT 11:14]