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\title{Math 10860, Honors Calculus 2}

\author{Homework 10~~~~~~~~~~~~~~~~{\bf NAME}:}

\date{Due by 11pm Tuesday April 28}

\maketitle

\subsection*{Instructions}

Yadda yadda yadda. Email to {\tt math10860homework at gmail.com}.

\subsection*{Reading}          
 
Class notes, Sections 15.4 (on Bolzano-Weierstrass) and Chapter 16 (on series). 
  
\subsection*{Assignment}   

\begin{enumerate}

%======================
\item Use the Bolzano-Weierstrass theorem (every bounded sequence has a convegent subsequence) to prove the first part of the Extreme Value Theorem: if $f:[a,b]\rightarrow {\mathbb R}$ is continuous, then there is $M$ such that $f(x) \leq M$ for all $x \in [a,b]$. ({\bf Hint}: Try a proof by contradiction.)
%======================

%======================
\item Decide whether the following sums converge. Explain your reasoning (i.e., which tests you are using, and why they apply.)
\begin{itemize}
\item $\sum_{n=1}^\infty (-1)^n\frac{\log n}{n}$. 
\item $\sum_{n=1}^\infty \frac{n^2}{n!}$. 
\item $\sum_{n=1}^\infty \frac{1}{(\log n)^n}$. 
\item $\sum_{n=2}^\infty \frac{1}{n(\log n)^2}$. 
\item $\sum_{n=1}^\infty \frac{1}{n^{1+1/n}}$.
\end{itemize}
%======================

%======================
\item 
\begin{enumerate}
\item In the sum below, $a$ is positive. Use the ratio test to decide for which values of $a$ the sum converges, and for which values it diverges: 
$$
\sum_{n=1}^\infty \frac{a^n n!}{n^n}.
$$
\item You should find that the ratio test gives no information at $a=e$ (if you didn't: redo part (a)!). When $a=e$, show that the series diverges, by using a result from the last  homework.
\item Decide when 
$$
\sum_{n=1}^\infty \frac{n^n}{a^n n!}
$$
converges, again using a result from the last  homework when the ratio test fails.
\end{enumerate}
%======================

%======================
\item Leibniz' alternating series test says that  if $(a_n)$ is a non-increasing sequence of non-negative numbers, and $(a_n)\rightarrow 0$ as $n \rightarrow 0$, then $\sum_{n=1}^\infty (-1)^n/n$ is finite. 

Is the hypothesis ``non-increasing'' necessary, or is the conclusion still valid if we merely assume that non-negative $a_n$ tends to $0$? 
%======================

%======================
\item
\begin{enumerate}
\item Prove that if $a_n \geq 0$ and $(a_n)$ is not summable (i.e., $\sum a_n$ diverges), then $(a_n/(1+a_n))$ is not summable.
\item Is the converse true? If $(a_n/(1+a_n))$ is not summable (with $a_n>0$), must it always be the case that $(a_n)$ is not summable?
\end{enumerate}
%======================

%======================
\item Wally, a slow but persistent worm, starts at
one end of a meter-long rubber band and crawls one centimeter per minute
toward the other end. At the end of each minute Wally rests for a moment. During that moment Karl, the equally persistent keeper of the band (whose sole purpose in life is to frustrate Wally) stretches the band one
meter. Thus after one minute of crawling, Wally is 1 centimeter from the start
and 99 from the finish; but in the moment that Wally rests, Karl stretches the band one meter. During the stretching
Wally maintains his relative position, 1\% from the start and 99\% from the finish. So at the end of Wally's moment of rest, he is 2cm from the starting point and 198cm from his
goal. After Wally crawls for another minute the score is 3cm traveled and 197
to go; but then Karl stretches one more meter (from 2 meters to 3 meters), and Wally's distances become 4.5cm travelled, and 295.5cm to go. And so on.
\begin{center}
Does Wally ever get to the end of the band???
\end{center}
He keeps moving, but the goal seems to
be moving away from him, faster than he moves. (We're assuming here infinite longevity for Karl and Wally,
infinite elasticity of the band, and an infinitely tiny worm.)
%======================


%======================
\item Define the {\em $7$-depleted harmonic number} $H^{(7)}_n$ to be the sum of the reciprocals of the natural numbers from $1$ to $n$, {\em except} those $n$ that have a $7$ in their decimal expansion. For example, $H^{(7)}_8 = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8}$. (There is no standard name or notation for this number).

Does $(H^{(7)}_n)_{n=1}^\infty$ converge or diverge?
%======================

%======================
\item 
\begin{enumerate}
\item Suppose that $(a_n)_{n=1}^\infty$ is weakly decreasing, with $a_n \geq 0$, and that $\sum_{n=1}^\infty a_n$ is finite. The vanishing condition says that $\lim_{n \rightarrow \infty} a_n =0$. Prove something stronger: $\lim_{n \rightarrow \infty} na_n =0$.
\item For each $\alpha > 0$, give an example of a sequence $(a_n)_{n=1}^\infty$ that is weakly decreasing, with $a_n \geq 0$, with $\sum_{n=1}^\infty a_n$ is finite, but with $\lim_{n \rightarrow \infty} n^{1+\alpha}a_n =+\infty$ (so, the result you proved in part (a) can't be improved upon).
\item Is the hypothesis ``weakly decreasing'' necessary?
\end{enumerate}
%======================

\end{enumerate} 

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