\section{Introduction}

\definition{}
Say we have a sequence $a_0, a_1, a_2, ...$. \par
The \textit{generating function} of this sequence is defined as follows:
\begin{equation*}
	A(x) = \sum_{n=0}^\infty a_nx^n = a_0 + a_1x + a_2x^2 + a_3x^3 + ...
\end{equation*}

Under some circumstances, this sum does not converge, and thus $A(x)$ is undefined. \par
However, we can still manipulate this infinite sum to get useful results even if $A(x)$
diverges.

\problem{}
Let $A(x)$ be the generating function of the sequence $a_n$, \par
and let $B(x)$ be the generating function of the sequence $b_n$. \par
Find the sequences that correspond to the following generating functions:
\begin{itemize}[itemsep=2mm]
	\item $cA(x)$
	\item $xA(x)$
	\item $A(x) + B(x)$
	\item $A(x)B(x)$
\end{itemize}

\begin{solution}
	\begin{itemize}[itemsep=2mm]
		\item $cA(x)$ corresponds to $ca_n$
		\item $xA(x)$ corresponds to $0, a_0, a_1, ...$
		\item $A(x) + B(x)$ corresponds to $a_n+b_n$
		\item $A(x)B(x)$ is $a_0b_0 + (a_0b_1 + a_1b_0)x + (a_0b_2 + a_1b_1 + a_2b_0)x^2 + ...$ \par
		Which corresponds to $c_n = \sum_{k=0}^n a_kb_{n-k}$
	\end{itemize}
\end{solution}


\vfill
\pagebreak




\problem{}<xminusone>
Assuming $|x| < 1$, show that
\begin{equation*}
	\frac{1}{1-x} = 1 + x + x^2 + x^3 + ...
\end{equation*}
\hint{use some clever algebra. What is $x \times (1 + x + x^2 + ...)$? }

\begin{solution}
	Let $S = 1 + x + x^2 + ...$ \par
	Then, $xS = x + x^2 + x^3 + ...$ \par

	\vspace{2mm}

	So, $xS = S - 1$ \par
	and $1 = S - xS = S(1 - x)$ \par
	and $S = \frac{1}{1-x}$.
\end{solution}


\vfill




\problem{}
Let $A(x)$ be the generating function of the sequence $a_n$. \par
Find the sequence that corresponds to the generating function $\frac{A(x)}{1-x}$

\begin{solution}
	\begin{align*}
		\frac{A(x)}{1-x}
		&=~ A(x)(1 + x + x^2 + ...) \\
		&=~ (a_0 + a_1x + a_2x^2 + ...)(1 + x + x^2 + ...)\\
		&=~ a_0 + (a_0 + a_1)x + (a_0 + a_1 + a_2)x^2 + ...
	\end{align*}

	Which corresponds to the sequence $c_n = \sum_{k=0}^n a_k$
\end{solution}

\vfill

\problem{}
Find short expressions for the generating functions for the following sequences:
\begin{itemize}
	\item $1, 0, 1, 0, ...$
	\item $1, 2, 4, 8, 16, ...$
	\item $1, 2, 3, 4, 5, ...$
\end{itemize}

\begin{solution}
	\begin{itemize}[itemsep=2mm]
		\item $1, 0, 1, 0, ...$ corresponds to $1 + x^2 + x^4 + ...$. \par
		By \ref{xminusone}, this is $\frac{1}{1-x^2}$.

		\item $1, 2, 4, 8, 16, ...$ corresponds to $1 + 2x + (2x)^2 + ...$. \par
		By \ref{xminusone}, this is $\frac{1}{1-2x}$.

		\item $1, 2, 3, 4, 5, ...$ corresponds to $1 + 2x + 3x^2 + 4x^3 + ...$.\par
		This is equal to $(1 + x + x^2 + ...)^2$, and thus is $\left(\frac{1}{1-x}\right)^2$
	\end{itemize}
\end{solution}


\vfill

\pagebreak
