Knowledge needed to understand generating functions

Question

I plan to take an undergraduate combinatorics course at my university, and I've seen generating functions as part of the curriculum. I am not a math student but I've seen some pure math courses outside the realm of calculus. My calculus knowledge has partially dissipated over time. I would like to know how much should I know about power series. Is knowledge about summations properties sufficient? I'll appreciate any other helpful advice.

Answer 1

Maybe knowing Newton's binomial theorem can help.

The binomial theorem that you learn in high school says $$ (x+y)^n = \sum_{k=0}^n \binom n k x^k y^{n-k} $$ for $n\in\{0,1,2,3,\ldots\}.$ Here we have \begin{align} \binom n k = {} & \frac{n!}{k!(n-k)!} \tag 1 \\[8pt] = {} & \frac{n(n-1)(n-2)\cdots(n-k+1)}{k!}. \tag 2 \end{align} Both lines $(1)$ and $(2)$ are correct, but line $(2)$ makes sense if $n = 3.14159$ or $n=-3.$ And in particular, if $n$ is a positive integer that is less than $k,$ then $\binom n k=0,$ so that we can write $$ (x+y)^n = \sum_{k=0}^\infty \binom n k x^k y^{n-k} $$ (i.e. with $\sum_{k=0}^\infty$ in place of $\sum_{k=0}^n$) provided $|y/x|<1.$ If $|y/x|\ge 1$ then the series does not converge.

Now notice that in combinatorics you will work with infinite series like this without worrying about whether they converge or not.

A series like this is used in proving that "negative sets are multisets." Multisets of size $5$ of a set $\{a,b,c\}$ of size $3$ are: $$ \{a,a,a,a,a\}, \{a,a,a,a,b\}, \{a,a,b,c,c\}, \text{ etc.} $$ The number of multisets of size $5$ of a set of size $3$ is $$ \left|\binom {-3}5\right| = \left| \frac{(-3)(-4)(-5)(-6)(-7)}{5\cdot4\cdot3\cdot2\cdot1} \right| = 21. $$ Recall that with $n\ge0$ and $n$ an integer, we have, for example $$ \binom 8 3 = \text{the number of subsets of size 3 of a set of size 8.} $$