Combinatorics

Formally: The generating function of an infinite sequence $(a_0, a_1, \ldots)$ is represented by the power series: $$ \sum_{n=0}^\infty a_nx^n $$ Informally: Consider a sequence of numbers $(a_0, a_1, \ldots)$. We want to encode this sequence within a function, expressed as: $$ a_0x^0 + a_1x^1 + a_2x^2 + \ldots + a_nx^n = \displaystyle\sum_{n=0}^\infty a_nx^n $$ Example of Sequence $(1, 1, 1, 1, \ldots )$: $$ 1x^0 + 1x^1 + 1x^2 + \ldots + 1x^n = \displaystyle\sum_{n=0}^\infty x^n = \frac{1}{1 - x} $$ You may ask why the equation $\displaystyle\sum_{n=0}^\infty x^n = \frac{1}{1 - x}$ holds....