Ok, I'm a bit stuck and need some help in the thinking process:
The reason I used the binomial form was to exploit the Maclaurin polynomial sequence approximation in simplifying the expression of E(N).
This is how I was thinking of it:
But the presence of the term (i-j)! in the denominator led me to decompose the summation in the end into three summations (I'll post later on), two of which can be simplified with Macluarin's approximation. The third has a capability of turning into a Maclaurin approximation form, but I can't seem to set it right.