108 lines
2.7 KiB
TeX
Executable File
108 lines
2.7 KiB
TeX
Executable File
\section{Introduction}
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\definition{}
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Say we have a sequence $a_0, a_1, a_2, ...$. \par
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The \textit{generating function} of this sequence is defined as follows:
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\begin{equation*}
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A(x) = \sum_{n=0}^\infty a_nx^n = a_0 + a_1x + a_2x^2 + a_3x^3 + ...
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\end{equation*}
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Under some circumstances, this sum does not converge, and thus $A(x)$ is undefined. \par
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However, we can still manipulate this infinite sum to get useful results even if $A(x)$
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diverges.
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\problem{}
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Let $A(x)$ be the generating function of the sequence $a_n$, \par
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and let $B(x)$ be the generating function of the sequence $b_n$. \par
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Find the sequences that correspond to the following generating functions:
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\begin{itemize}[itemsep=2mm]
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\item $cA(x)$
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\item $xA(x)$
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\item $A(x) + B(x)$
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\item $A(x)B(x)$
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\end{itemize}
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\begin{solution}
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\begin{itemize}[itemsep=2mm]
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\item $cA(x)$ corresponds to $ca_n$
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\item $xA(x)$ corresponds to $0, a_0, a_1, ...$
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\item $A(x) + B(x)$ corresponds to $a_n+b_n$
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\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
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Which corresponds to $c_n = \sum_{k=0}^n a_kb_{n-k}$
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\end{itemize}
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\end{solution}
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\vfill
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\pagebreak
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\problem{}<xminusone>
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Assuming $|x| < 1$, show that
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\begin{equation*}
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\frac{1}{1-x} = 1 + x + x^2 + x^3 + ...
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\end{equation*}
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\hint{use some clever algebra. What is $x \times (1 + x + x^2 + ...)$? }
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\begin{solution}
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Let $S = 1 + x + x^2 + ...$ \par
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Then, $xS = x + x^2 + x^3 + ...$ \par
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\vspace{2mm}
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So, $xS = S - 1$ \par
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and $1 = S - xS = S(1 - x)$ \par
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and $S = \frac{1}{1-x}$.
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\end{solution}
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\vfill
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\problem{}
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Let $A(x)$ be the generating function of the sequence $a_n$. \par
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Find the sequence that corresponds to the generating function $\frac{A(x)}{1-x}$
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\begin{solution}
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\begin{align*}
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\frac{A(x)}{1-x}
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&=~ A(x)(1 + x + x^2 + ...) \\
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&=~ (a_0 + a_1x + a_2x^2 + ...)(1 + x + x^2 + ...)\\
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&=~ a_0 + (a_0 + a_1)x + (a_0 + a_1 + a_2)x^2 + ...
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\end{align*}
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Which corresponds to the sequence $c_n = \sum_{k=0}^n a_k$
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\end{solution}
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\vfill
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\problem{}
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Find short expressions for the generating functions for the following sequences:
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\begin{itemize}
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\item $1, 0, 1, 0, ...$
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\item $1, 2, 4, 8, 16, ...$
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\item $1, 2, 3, 4, 5, ...$
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\end{itemize}
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\begin{solution}
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\begin{itemize}[itemsep=2mm]
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\item $1, 0, 1, 0, ...$ corresponds to $1 + x^2 + x^4 + ...$. \par
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By \ref{xminusone}, this is $\frac{1}{1-x^2}$.
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\item $1, 2, 4, 8, 16, ...$ corresponds to $1 + 2x + (2x)^2 + ...$. \par
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By \ref{xminusone}, this is $\frac{1}{1-2x}$.
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\item $1, 2, 3, 4, 5, ...$ corresponds to $1 + 2x + 3x^2 + 4x^3 + ...$.\par
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This is equal to $(1 + x + x^2 + ...)^2$, and thus is $\left(\frac{1}{1-x}\right)^2$
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\end{itemize}
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\end{solution}
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\vfill
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\pagebreak
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