I'm learning about generating functions and in the opening explanations my book (and various sources) claim:
$$a_n = 1 \forall n \in \mathbb{N}_0, \ \ \ f(x) = \frac{1}{1-x}$$.
I read this as:
The sum of the $nth$ term of a series where every term is $1$ is equal to $$\frac{1}{1-n}$$.
But this doesn't make sense. Look at the 5th term: $$1+1+1+1+1 = 5$$ but this is not consistent with the formula: $$\frac{1}{1-5} = \frac{1}{-4}$$
What am I misunderstanding?
On
All they are saying is the well-known Taylor expansion of the geometric series:
$$ \frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + x^5 + \cdots $$
Note that the coefficient of each power of $x$ is 1. That is, if we write
$$ \frac{1}{1-x} = a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots = \sum_{n=0}^\infty a_nx^n $$
then $a_n = 1$ for all $n$.
The definition of an (ordinary) generating function is that if $f(x)$ generates the sequence $a_n$, then: $$a_n = f^{(n)}(0)/n!\ .$$
Here we have $$f(x) = \frac{1}{1-x} = \sum_{n=0}^\infty 1\cdot x^n = \sum_{n=0}^\infty a_n x^n$$ where $a_n=1$ for all $n$.