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?
Are there any simple excercises about this topic?
Tomorrow I have an exam, and I'm not able to understand anything about this, even though I've read a couple of websites I've found about generating functions.
Sorry if I've mistaken some math words, I haven't been taught math in english.