# Why is it important to have the closed form of a generating function?

I am having introductory lectures on combinatorial analysis, I've been presented to the concept of generating functions and it's applications to solving combinatorial problems. The generating function of the sequence $(1,1,1,1,1,1,\ldots)$ is:

$$1+x+x^2+x^3+x^4+x^5+\ldots\tag{1}$$

Which can be reduce to a closed form expression:

$$\displaystyle\frac{1}{1-x}\tag{2}$$

But until the present moment, the exercises I made in the chapters we're studying asked us to write $(1)$ as $(2)$, there wasn't anything about using the closed form to obtain further results. In the combinatorial exercises I made, I had only to evaluate the coefficient of a certain term in the generating function or the exponent of a certain coefficient in the formal series and at least in the book I'm using, there is still no use to the closed form of certain formal power series.

So why is it important to have a closed form such as $(2)$? I still don't get why it is important/useful.

• you have two sequences (and it's gfs) - A and B and you'd like to convolute them (also known as Cauchy product). Multiple this gfs closed forms and you're done - now you can convert it to first form. For more check: math.upenn.edu/~wilf/DownldGF.html
– fex
Jan 6, 2015 at 13:24