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\date{26 March 2018}
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\begin{frame}{Pointwise and uniform convergence}
\textbf{Definition} (``pointwise convergence"): Let $f_n \colon [a,b] \rightarrow \mathbb{R}$ be a sequence of functions. We say that $f_n$ converges pointwise to $f$ provided that $\forall x \in[a,b] \forall \epsilon>0 \exists N \in \mathbb{N}$ so that for all $n \geq N, |f_n(x)-f(x)|<\epsilon$. For this, we write $f_n \rightarrow f$. 

\textbf{Definition} (``uniform convergence"): Let $f_n \colon [a,b] \rightarrow \mathbb{R}$ be a sequence of functions. We say that $f_n$ converges uniformly to $f$ provided that $\forall \epsilon > 0 \exists N \in \mathbb{N}$ so that for all $n \geq N$ and $x \in [a,b]$, $|f_n(x)-f(x)| < \epsilon$. For this, we write $f_n \rightrightarrows f$.
\end{frame}


\begin{frame}{Four questions}
Let $f_n \colon [a,b] \rightarrow \mathbb{R}$ be a sequence of functions. Let define the function 
$$F(x)=\displaystyle\sum_{k=0}^{\infty} f_n(x),$$
and assume that the series always converges (so this definition makes sense). 

\begin{enumerate}[1.]
\item What conditions guarantee that $F$ is continuous?
\item What conditions guarantee that $F$ is differentiable?
\item What conditions allow us to write
$$\displaystyle\int_a^b F(x) \mathrm{d}x = \displaystyle\int_a^b f_1(x) \mathrm{d}x + \displaystyle\int_a^b f_2(x) \mathrm{d}x + \ldots?$$
\item What conditions allow us to write $F'(x)=f_1'(x)+f_2'(x)+\ldots$?
\end{enumerate}
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\begin{frame}{From last class}
\textbf{Last Friday}: we argued that the sequence $f_n \colon [0,1] \rightarrow \mathbb{R}$ defined by $f_n(x)=x^n$ was pointwise convergent to zero (i.e. $f_n \rightarrow 0$) but not uniformly convergent to zero, (i.e. $\not\rightrightarrows 0$). 
\end{frame}

\begin{frame}{Four similar questions}
Let $\mathcal{F}(x) = \displaystyle\lim_{n \rightarrow \infty} f_n(x)$. 
\begin{enumerate}[1.']
\item What conditions on $f_n$ will guarantee that $\mathcal{F}$ is cotinuous?
\item What conditions on $f_n$ will guarantee that $\mathcal{F}$ is differentiable?
\item What conditions on $f_n$ will guarantee that 
$$\displaystyle\int_a^b \mathcal{F}(x) \mathrm{d}x = \displaystyle\lim_{n \rightarrow \infty} \displaystyle\int_a^b f_n(x) \mathrm{d}x?$$
\item What conditions on $f_n$ will guarantee that $\mathcal{F}'(x) = \displaystyle\lim_{n \rightarrow \infty} f_n'(x)$?
\end{enumerate}
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\begin{frame}{In-class problem}
Let $f_n \colon [0,1] \rightarrow \mathbb{R}$ be defined by $f_n(x) = \dfrac{x^n}{n}$ for $n=1,2,3,\ldots$. \\
What is $\mathcal{F}(x)$? \onslide<2->{\textbf{Answer}: $\mathcal{F}(x)=\displaystyle\lim_{n \rightarrow \infty} \dfrac{x^n}{n} = 0$.} \\
What is $\mathcal{F}'(x)$? \onslide<2->{\textbf{Answer}: $\mathcal{F}'(x)=0$.} \\
What is $f_n'(x)$? \onslide<2->{\textbf{Answer}: $f_n'(x)=x^{n-1}$.} \\
What is $\displaystyle\lim_{n \rightarrow \infty} f_n(x)$? \onslide<2->{\textbf{Answer}: $\displaystyle\lim_{n \rightarrow \infty} f_n'(x) = \left\{ \begin{array}{ll}
0, & \quad 0 \leq x < 1 \\
1, & \quad x=1.
\end{array} \right.$} \\
\onslide<2->{Hence: 
$$\mathcal{F}'(x) \neq \displaystyle\lim_{n \rightarrow \infty} f_n'(x).$$.}
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\begin{frame}{Examples}
Let $f_n \colon [0,1] \rightarrow \mathbb{R}$ be defined by
$$f_n(x) = \left\{ \begin{array}{ll}
4n^2x, & \quad 0 \leq x \leq \dfrac{1}{2n} \\
-4n^2x + 4n, & \quad \dfrac{1}{2n} \leq x \leq \dfrac{1}{n} \\
0, & \quad \dfrac{1}{n} \leq x \leq 1.
\end{array} \right.$$
Draw $f_n$ for some $n$ and compute $\displaystyle\int_0^1 f_n(x)$ for your value of $n$. \\
What is $\displaystyle\lim_{n \rightarrow \infty} f_n(x)$ for any value of $x$?
\end{frame}

\begin{frame}{References}
``Advanced Calculus" by A.E. Taylor
\end{frame}
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