00:01
In this question, we are asked to find the radius and the interval of convergence of the series.
00:05
And to do that, we will use the ratio test.
00:11
By the ratio test, we need to calculate the limit of the absolute value of a n plus one over a n as n goes to infinity, where a n is the general term of the series.
00:31
To get a n plus one, we simply need to replace n by n plus one.
00:35
We will get x to the n plus third power divided by three multiplied by n plus one factorial.
00:44
And this is factorial.
00:46
And then we need to divide that by a n, which is x to the n plus two divided by three times n factorial.
01:04
It's important to note here that three is not inside the factorial, right? three is outside.
01:17
We can rewrite that as the limit of x to the n plus three divided by three times n plus one factorial multiplied by three n factorial divided by x to the n plus two.
01:32
We can cancel three and we can cancel x to the n plus second power.
01:46
We will get the limit of the absolute value of x.
01:49
And we can rewrite n factorial as one times two times so on times n...