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\begin{center}
\textbf{\large Math 148 Assignment 5}\\[1ex]
\textbf{Due  2:00 p.m.\ Friday, March 13 in the Math 148 dropbox.}  
\end{center}

\vskip 2em 
\begin{Ex}

\item Let $f_n(x) = xne^{-nx}$ for  $x \ge 0$ and $n \ge 1$.
Find $\dlim_{n\to\infty} f_n(x)$. Is this limit uniform on $[0,\infty)$?


\item Let $f_n(x) = \dfrac x{1+nx^2}$ for $x \in \bR$ and $n \ge 1$.
Find $\dlim_{n\to\infty} f_n(x)$. Is this limit uniform on $\bR$?

\item Let $f_n$ and $g_n$ be continuous functions on  $[a,b]$ for $n \ge1$.
 Suppose that $f_n$ converges uniformly to $f$ and $g_n$ converges
 uniformly to $g$ on  $[a,b]$.
 Prove that $f_ng_n$ converges uniformly to $fg$ on $[a,b]$.


\item  Let $f_n(x) = \dfrac{x^2} {(1+x^2)^n}$ for $x\in\bR$, and let 
$s_k(x) = \dsum_{n=0}^k f_n(x)$. 
\begin{parts}
\item Find $\dlim_{k\to\infty} s_k(x)$.
\item For which values $a<b$ does this series converge uniformly on $[a,b]$?
\end{parts}


\item  For $n \ge 1$, define functions $f_n$ on $[0,\infty)$ by
\\[1.ex] $\displaystyle\strut\qquad
 f_n (x) =
 \begin{cases}
  e^{-x} &\qfor 0 \le x \le n\\
  e^{-2n} (e^n + n - x) &\qfor n \le x \le n + e^n \\
  0 & \qfor x \ge n + e^n.
 \end{cases} 
 $\\
\begin{parts}
 \item Find the pointwise limit $f$ of $f_n$.  Show that the
 convergence is uniform on $[0,\infty)$.

 \item Compute $\dint_0^\infty f(x) \,dx$ and
 $\dlim_{n\to\infty} \dint_0^\infty f_n(x) \,dx$.

 \item Why does this not contradict Integral Convergence Theorem?
\end{parts}


\item 
\begin{parts}
\item Suppose that $f : \bR \to \bR$ is uniformly continuous.  
Let $f_n(x) = f(x+1/n)$.\\
Prove that $f_n$ converges uniformly to $f$ on $\bR$.
\item Does this remain true if $f$ is just continuous?  
Prove it or provide a counterexample.
\end{parts} 

\item For which values of $x\ge 1$ does the expression
$x^{x^{x^{x^{.^{.^{.}}}}}}$ make sense?  \\
To tackle this, define  $f_1(x)=x$ and $f_{n+1}(x)=x^{f_n(x)}$ for $x \ge 1$ and $n\ge1$.  
\begin{parts}
\item Show that $f_{n+1}(x) \ge f_n(x)$ for all $n\ge1$.  
\item When $L(x) = \dlim_{n\to\infty} f_n(x)$ exists, find an equation for $L(x)$.\\
Use it to find an upper bound for $x$.
\item For these values of $x$, show by induction that
 $f_n(x)$ is bounded above by $e$ for all $n\ge1$.  
 What can you conclude?
\item What happens for larger $x$?
\end{parts}

\end{Ex}
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