00:01
In this question, we are asked to find the radius of convergence and the interval of convergence of the given series.
00:07
And for that, we will use the ratio test.
00:13
By the ratio test, we need to calculate the limit of the absolute value of a n plus 1 over a n as n goes to infinity, where a n is the general term of the series.
00:32
We can ignore the negative sign, the negative 1 to the n because it gets cancelled by the absolute value sign.
00:38
And to get a n plus 1, we need to replace n by n plus 1.
00:49
So this is a formula for a n plus 1, and we need to divide that by a n.
01:04
By the division of fractions rules, we will get the limit of the absolute value of x to the n plus first, divided by the square root of n plus 5, multiplied by the square root of n plus 4, divided by x to the n.
01:22
We can cancel x to the n.
01:26
We will get the limit of the absolute value of x multiplied by the square root of n plus 4 over n plus 5.
01:39
When n goes to infinity, n plus 4 over n plus 5 goes to 1, and this equals to the absolute value of x.
01:52
By the ratio test, we want that to be less than 1 for the series to converge.
01:58
And that means that the radius of convergence equals to 1.
02:05
However, the ratio test doesn't tell us what happens at the endpoints of this interval, when the limit equals to 1.
02:13
We have to check that separately.
02:18
The absolute value of x equals 1 means either x equals to negative 1 or x equals to 1...