Mark/handouts
Mark/handouts
1
0
Fork 0
Code Pull Requests 2 Actions Packages Releases Activity

Typos

Browse Source
This commit is contained in:
Mark 2025-01-16 16:49:45 -08:00
parent fb41e67a8f
commit b236a3b271
Signed by: Mark
GPG Key ID: C6D63995FE72FD80
2 changed files with 13 additions and 12 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

3
Advanced/Generating Functions/parts/00 introduction.tex
View File

@ -4,7 +4,7 @@
Say we have a sequence $a_0, a_1, a_2, ...$. \par Say we have a sequence $a_0, a_1, a_2, ...$. \par
The \textit{generating function} of this sequence is defined as follows: The \textit{generating function} of this sequence is defined as follows:
\begin{equation*} \begin{equation*}
A(x) = \sum_{n=0}^\infty a_nx^n = a_0 + a_1x + a_2x^2 + a_3x+3 + ... A(x) = \sum_{n=0}^\infty a_nx^n = a_0 + a_1x + a_2x^2 + a_3x^3 + ...
\end{equation*} \end{equation*}
Under some circumstances, this sum does not converge, and thus $A(x)$ is undefined. \par Under some circumstances, this sum does not converge, and thus $A(x)$ is undefined. \par
@ -44,6 +44,7 @@ Assuming $|x| < 1$, show that
\begin{equation*} \begin{equation*}
\frac{1}{1-x} = 1 + x + x^2 + x^3 + ... \frac{1}{1-x} = 1 + x + x^2 + x^3 + ...
\end{equation*} \end{equation*}
\hint{use some clever algebra. What is $x \times (1 + x + x^2 + ...)$? }
\begin{solution} \begin{solution}
Let $S = 1 + x + x^2 + ...$ \par Let $S = 1 + x + x^2 + ...$ \par

22
Advanced/Generating Functions/parts/01 fibonacci.tex
View File

@ -50,7 +50,7 @@ we used to define the Fibonacci numbers.
\pagebreak \pagebreak
\problem{} \problem{}<fibo>
Using the problems on the previous page, find $F(x)$ in terms of $x$. Using the problems on the previous page, find $F(x)$ in terms of $x$.
\begin{solution} \begin{solution}
@ -77,7 +77,7 @@ A \textit{rational function} $f$ is a function that can be written as a quotient
That is, $f(x) = \frac{p(x)}{q(x)}$ where $p$ and $q$ are polynomials. That is, $f(x) = \frac{p(x)}{q(x)}$ where $p$ and $q$ are polynomials.
\problem{} \problem{}
Solve this equation for $F(x)$, expressing it as a rational function. Solve the equation from \ref<fibo> for $F(x)$, expressing it as a rational function.
\begin{solution} \begin{solution}
\begin{align*} \begin{align*}
@ -98,20 +98,20 @@ Solve this equation for $F(x)$, expressing it as a rational function.
\pagebreak \pagebreak
\problem{}<pfd> \definition{}
Now that we have a rational function for $F(x)$, \par \textit{Partial fraction decomposition} is an algebreic technique that works as follows: \par
find a closed-form expression for its coefficients. If $p(x)$ is a polynomial and $a$ and $b$ are constants,
we can rewrite the rational function $\frac{p(x)}{(x-a)(x-b)}$ as follows:
\vspace{2mm}
Do this using \textit{partial fraction decomposition:} \par
We can break up a rational function $\frac{p(x)}{(x-a)(x-b)}$ as follows:
\begin{equation*} \begin{equation*}
F(x) = \frac{p(x)}{(x-a)(x-b)} = \frac{c}{x-a} + \frac{d}{x-b} \frac{p(x)}{(x-a)(x-b)} = \frac{c}{x-a} + \frac{d}{x-b}
\end{equation*} \end{equation*}
where $c$ and $d$ are constants. where $c$ and $d$ are constants.
\problem{}<pfd>
Now that we have a rational function for $F(x)$, \par
find a closed-form expression for its coefficients using partial fraction decomposition.
\begin{solution} \begin{solution}
\begin{align*} \begin{align*}
F(x) F(x)