00:01
Okay, we want to test an infinite series for convergence.
00:03
First one is this one.
00:05
Let's do the limit test of just the thing inside the sum.
00:22
That goes to 1 1 2.
00:25
You divide numerator and denominator by n, and then let n go to infinity, you get 1 1 2.
00:31
That is not equal to zero, therefore the sum diverges.
00:47
Okay, so for this one, this one we use the limit comparison test.
00:57
Okay, and the sum we're interested in then is the sum of one over n to the 3 1 2 power, which by the way is a convergent series because of the p -test.
01:12
So let's look at this.
01:34
Okay, then i do complex fractions on that, and i get, okay, so if i divide numerator and denominator by n cubed, this is what i get.
02:29
All right, that's not zero or infinity, so therefore the sum converges since the sum of one over n to the 3 1 2 does, part c.
02:57
For this one, i'm gonna use the integral test.
03:02
So we have to evaluate this integral.
03:14
I'll make a substitution, so this becomes integral one to infinity of 1 1 2.
03:45
That is finite, and that means that the sum converges.
04:29
Okay, so for this one, i'm gonna use the alternating series test.
04:37
So it's definitely alternating.
04:47
Part two is we need to show that it's decreasing.
05:01
So that means that if we take that ratio of successive terms, there's a better way to do it than that...