00:01
Okay, so the divergence test says that if the limit as n tends to infinity of a .n is infinite, or is, actually, no, it's just non -zero, then the sum of a .n as a series diverges.
00:30
So for our series in series a, we have that a -n is equal to 3n plus 2 squared over a number.
00:38
4 n squared minus 9.
00:41
And so we can see that the limit as n tends to infinity of a n is equal to sorry, that should be n, that should be a square.
00:51
If we divide the top and the bottom of a n by squared, which we're allowed to do, because if you divide the fraction on the top and the bottom by the same thing, then the fraction stays the same, then this is equal to 3 plus 2 over n squared, divided by 4 minus 9 over n squared.
01:10
And as n tends to infinity, we have that 2 over n tends to 0, and 9 over n squared tends to 0.
01:19
And so we get that the limit as n tends to infinity is 3 squared over 4 or 9 over 4.
01:26
This is clearly non -zero, and so the series diverges.
01:35
For part b, to show the series is alternating, all we need is that for n, odd, the series is minus 2 over 3n.
01:45
For n even, the series is plus 2 over 3n.
01:50
And so we can see there's a sign change and therefore it's alternating.
01:59
For part 2, we're then asked to use the alternating series test to state conclusions about the limit of the series...