00:01
We're given a sequence.
00:02
We're asked to determine the sequence is monotonic, and if it is bounded, sequence is a .n equals 2dn, 3dn over in factorial.
00:16
So notice this is the same as the exponent laws 2 times 3 or 6 bn over in factorial.
00:33
And we have by theorem 5 from the book, the limit that's end approaches infinity, of x bn over n factorial equal to zero for any real number x and so taking x to be six this shows the limit as an approach of infinity of a .m is going to be zero and that means that sequence a .n converges and we have therefore since it converges that sequence a .m.
01:32
Is found however, we calculate the same values of sequence.
01:59
So we have the a1 is going to be 6 over 1 factorial.
02:08
So simply 6 .a2 is going to be 6 squared over 2 factorial, which is 36 over 2 or 18.
02:27
A3 is equal to 6 cubed over 3 factorial, which is 3.
02:33
Is 6 cubed over 3 times 2 can cancel out that to get 6 squared, which is simply 36.
02:44
So far the sequence is increasing.
02:49
A4 is 6 to the 4 over 4 factorial, which is the same as 6 cubed over 4 times 2, and not even 2, which is 4.
03:03
And this can also be written as 3 cubed times 2, which is 8 times 2, which is 16.
03:18
And so we have that a3 is greater than a2, but a3 is also greater than a4...