# Ordinary generating functions - I can't understand this

I'm trying to understand ordinary generating functions. I've been looking for any tutorial or some explanations about the topic but I haven't found anything useful and - what's more important - well explained.

1) Do ALL ordinary generating functions look like the polynomial 1 + a1x + a2x^2 + a3x^3 + ..., where (a)n depend on the given sequence? So for example the generating function for the (1, 0, 1, 0, ...) sequence would be 1 + a1x + a3x^3 + a5x^5 + ... ?

2) What actually is the generating function? What's the point of adding increasing powers of x to the sequence elements and summing them up?

Sorry if I've mistaken some math words, I haven't been taught math in english.

The ordinary generating function for the sequence $a_0,a_1,a_2,\dots$ is $a_0+a_1x+a_2x^2+\cdots$. So the OGF for $1,0,1,0,\dots$ is $1+x^2+x^4+\cdots$.