Edit notes while doing DCT gamma function question

analysis--berkeley-202a.tex

@@ -925,6 +925,8 @@ \subsection{For a sequence of sets}
 
 We have $\liminfn A_n \subseteq \limsupn A_n$.
 
+\url{https://math.stackexchange.com/a/476171/397805}
+
 
 \section*{Expressions involving countable unions and intersections}
 

analysis--real-analysis-sequences-series.tex

@@ -276,11 +276,24 @@ \subsubsection{Maclaurin series derivation}
 
 and in general $c_n = \frac{f^{(n)}(0)}{n!}$.
 
-For example, take $f(x) = e^x$. The derivatives are all the same of course: $f^{(n)}(x) = e^x$. And $e^0 = 1$, so
+For example, take $f(x) = e^x$. The derivatives are all the same of course: $f^{(n)}(x) = e^x$. And $e^0 = 1$, so the Maclaurin expansion of $e^{x}$ is
+\begin{align*}
+  e^x = e^{0} + \frac{xe^0}{1!} + \frac{x^2e^0}{2!} + \cdots = \sum_{n=0}^\infty \frac{x^n}{n!}
+\end{align*}
+The Maclaurin expansion of $e^{-x}$ is
+\begin{align*}
+  e^{-x} = e^{0} - \frac{xe^0}{1!} + \frac{x^2e^0}{2!} - \cdots = \sum_{n=0}^\infty (-1)^{n}\frac{x^n}{n!}.
+\end{align*}
 
 \begin{align*}
-  e^x = 1 + \frac{x}{1} + \frac{x^2}{2!} + \frac{x^3}{3!} + ... = \sum_{i=0}^\infty \frac{x^i}{i!}
+  \lim_{n \to \infty} \(1 + \frac{x}{n}\)^n
+  &= \lim_{n \to \infty} \sum_{k=0}^n {n \choose k} \(\frac{x}{n}\)^k \\
+  &= \lim_{n \to \infty} \sum_{k=0}^n \frac{n!}{(n-k)!k!} \(\frac{x}{n}\)^k \\
 \end{align*}
+... \red{TODO} prove this is $e^x$.
+
+
+
 
 \section{Exercises}
 

calculus.tex

@@ -221,9 +221,9 @@ \subsection{Integration by parts}
 
 
 \begin{theorem}
-  Let $f:U \to U$ and $g:U \to U$. Then
+  Let $f:X \to X$ and $g:X \to X$. Then
   \begin{align*}
-    \int fg' \du = fg - \int gf' \du.
+    \int fg' \dx = fg - \int gf' \dx.
   \end{align*}
 \end{theorem}
 

foundations.tex

@@ -79,7 +79,7 @@ \section{Permutations and combinations}
 \begin{theorem*}
   The number of $k$-tuples that can be formed from $\{1, 2, \ldots, n\}$ is
   \begin{align*}
-    P(n, k) = n_{(k)} = n(n-1)(n-2)\cdots(n-k+1).
+    P(n, k) = n_{(k)} = n(n-1)(n-2)\cdots(n-k+1) = \frac{n!}{(n - k)!}.
   \end{align*}
   The number of sets of size $k$ that can be formed from $\{1, 2, \ldots, n\}$ is
   \begin{align*}
@@ -92,6 +92,18 @@ \section{Binomial theorem}
   (a + b)^n = \sum_{k=0}^n{n \choose k}a^{n-k}b^k
 \end{align*}
 
+\section{Taylor expansions}
+
+Suppose that any function of a real number $f(x)$ can be represented by a "power series" with certain coefficients $c_i$
+\begin{align*}
+  f(x) = c_0 + c_1x^1 + c_2x^2 + c_3x^3 + c_4x^4 + ...
+\end{align*}
+Examining successive derivatives shows that $c_n = \frac{f^{(n)}(0)}{n!}$ (\red{TODO} explain why evaluating at zero).
+For example, the Maclaurin expansion of $e^{x}$ is
+\begin{align*}
+  e^x = e^{0} + \frac{xe^0}{1!} + \frac{x^2e^0}{2!} + \cdots = \sum_{n=0}^\infty \frac{x^n}{n!}
+\end{align*}
+
 \section{Triangle inequalities}
 
 \begin{theorem*}