00:01
We're given a sequence, and we're asked to determine if the sequence is monotonic and if it is bounded.
00:13
Sequences, a .n equals 2 minus 2 over m, minus 1 over 2 to the n.
00:27
Natural numbers tell us that 1 is less than equal to n is strictly less than n plus 1, and so it follows that 2 is going to be strictly less than or equal to 2n, which is strictly less than 2n plus 2.
00:52
Another way to think of this is what we have these are positive numbers and so if we exponentiate 2 we get the 2 the 1 less than 2 to the n plus 1 and therefore it follows that 0 is going to be less than we also have the 0 plus 1 over n plus 1 is less than 1 over n which is less than equal to 1 and so it follows that 0 is less than 1 over 2 to the n plus 1, which is less than 1 over 2 to the n, which is less than or equal to 1.
02:06
In fact, it's less than equal to 1 over 2 of the first, which is simply 1 half.
02:20
And so we have that negative 2 is less than or equal to negative 2 over n.
02:46
This is less than a negative 2 over n plus 1, which is less than 0...