00:01
For this problem, we are asked to state a variation of the ratio test from theorem 7 .6 that would allow one to use ratios to test a sequence ak for monotonicity when each ak is less than zero.
00:13
So first, let's think about monotonic increase.
00:19
So we know that if something's monotonically increasing, then we have that ak will always be less than ak plus 1.
00:27
Except, actually, let's see, yeah, in this case, that would still be the case, strictly speaking.
00:32
It's just that ak is always less than ak plus 1 and both of them are always less than 0.
00:38
If we want to express this for in the form of a ratio, we need to have then that ak over ak plus 1 would have to be less than 1, but because we've divided through by essentially a negative value, we know it's negative...