00:01
We are given a sequence and we are asked to find a closed form for the exponential generating function for the sequence.
00:09
In part a, the sequence is defined as a n equals 2 for all n.
00:20
And so we have the exponential generating function if it's g is equal to, by definition, the sum from k equals 0 to infinity of a sub k.
00:39
Over k factorial x to the k and this is equal to the sum from k equal zero to infinity of two over k factorial x to the k which is equal to two times the sum from k equal zero to infinity of x to the k over k factorial which by definition this is going to be or this is the taylor series of e to the x and so g of x is two e of the x of the x in part b, we're given the sequence defined as a .n equals negative 1 to the n.
01:30
So an alternating sequence.
01:35
And by the definition of the exponential generating function, g of x is equal to the sum from k equals 0 to infinity of a .n, which in this case is negative 1 to the n over k factorial, x to the k to the k.
01:56
And this is equal to the sum from k equals 0 to infinity of negative x to k over k factorial and by the taylor series expansion for e of the x this is equal to each of the negative x in part c we're given the sequence to find a n equals 3 to n by definition of x for generating function, g of x is equal to sum from k equals 0 to infinity of 3 to the k over k factorial x to the k, which is equal to the sum from k equals 0 to infinity of 3x to the k over k factorial, which by a taylor series expansion for e to the x is e to the 3x.
03:04
In part d, we're given a sequence, a .n equals n plus 1.
03:24
By definition, the exponential generating function g of x is equal to the sum from k equals 0 to infinity of k plus 1 over k factorial times x to the k...