Chapter 6: Additional Topics

There is an extremely powerful tool in discrete mathematics used to manipulate sequences called the generating function. The idea is this: instead of an infinite sequence (for example: $2,3,5,8,12,\dots$) we look at a single function which encodes the sequence. But not a function which gives the $n$th term as output. Instead, a function whose power series (like from calculus) “displays” the terms of the sequence. So for example, we would look at the power series $$2+3x+5x^2+8x^3+12x^4+\cdots$$ which displays the sequence $2,3,5,8,12,\dots$ as coefficients.

An infinite power series is simply an infinite sum of terms of the form $c_n{x}^n$ were $c_n$ is some constant. So we might write a power series like this: