Find E(Y^3) of Poisson, without using generating functions

Question.

Let Y be a random variable that has a Poisson distribution with parameter . Without using generating functions:

(a) show that E(X) =

(b) find E(X^3).

Answer

(a)

(I used exp expansion for summation)

(b) and now I am stuck with the powers of x in my summation expression...

I tried to substitute x-1=j:

and then possibly expand (j+1)^2 into {j^2+2j+1} but then I have to deal with j^2 again (as with x^2 before), so it does not seem to help.

Are there any methods to deal with these kinds of summation calculations? I have looked into many books but they all show calculations of n-th moments straight from the moment generating functions, and my examinor seem to prefer his students to calculate from 'first principles' - this is a sample exam question. Unfortunately, there is no answer to this question in the study guide.