00:01
All right, we want to find the radius and interval of convergence for this series.
00:05
So we're going to use the ratio test.
00:07
We're going to take the limit as in goes to infinity.
00:11
X plus 3 to the n plus 1 over 3 to the end plus 1, natural log of n plus 1, all over x plus 3 to the n, 3 to the n, natural log of n.
00:30
All right, so it's a four -story fraction.
00:32
So i'm going to flip and i'm going to put people with who they belong.
00:37
So x plus 3 to the n plus 1 over x plus 3 to the n over 3 to the n plus 1.
00:50
Ln of n over the ln of n of n plus 1.
00:57
All right.
00:58
Then by algebra, if you have n plus 1 of something over n of something, then you just have one of them left.
01:08
If you have 3 to the n over 3 to the n plus 1, you have a 3 on the bottom, and then that will have to work on.
01:21
So when you take the limit, the x plus 3 over 3 can come out here, and we have to take the limit as in goes to infinity of the natural log of n over the natural log of n plus 1.
01:40
All right, so that's infinity over infinity, so i'm going to know lopetal's rule.
01:46
1 over in 1 over n plus 1 4 story fraction invert so you m plus 1 over in so when you low pital's rule you get or when you plug in infinity you get infinity over infinity so then you lo petal's rule and you get 1 over 1 so that's 1 so this is x plus 3 over 3 less than 1 so you get hex absolute value of x plus 3 is less than 3.
02:33
So the radius of convergence is 3.
02:36
Sorry about that.
02:37
I don't get messages.
02:41
Okay.
02:41
So now we've got to go x plus 3 less than 3, but bigger than minus 3.
02:48
So negative 6 less than x less than 0.
02:53
So now we've got to check both of those.
02:58
If x equals 0, then the series turns into 3.
03:04
3 to the n over 3 to the n.
03:12
Natural log of n.
03:16
So 1 over the natural log of n.
03:21
Okay, now we have to decide if that's convergent or not.
03:25
All right, so here's what i know.
03:31
Here's the natural log of n.
03:36
Here's n...