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\begin{center}
\textbf{\large Math 148 Assignment 4}\\[1ex]
\textbf{Due  2:00 p.m.\ Monday, March 2 in the Math 148 dropbox.}  
\end{center}

\vskip 2em 
\begin{Ex}

\item Let $a_n$ be a monotone decreasing sequence of positive terms. \vspace{-2ex} Prove that
$\dsum_{n=1}^\infty a_n$ converges if and only if $\dsum_{n=0}^\infty 2^n a_{2^n}$ converges.\vspace{-1ex}



\item It is known that $\dsum_{n=1}^\infty \dfrac{1}{n^4}  = \dfrac{\pi^4}{90}$. 
Use this fact to compute $\dsum_{n=1}^\infty \dfrac{(-1)^n}{n^4}$.


\item Suppose that $a_n \le b_n \le c_n$ for all $n\ge 1$.
If $\dsum_{n=1}^\infty a_n$ and $\dsum_{n=1}^\infty c_n$ converge, 
does $\dsum_{n=1}^\infty b_n$ converge?
Prove it or provide a counterexample.


\item Suppose that $\dsum_{n=1}^\infty a_n = A$ exists and $(b_n)$ is a monotone sequence with limit $B$. \vspace{-1ex}
Prove that $\dsum_{n=1}^\infty a_nb_n$ converges.
\hint Find a way to apply Dirichlet's test.\\


\item  Decide which of the following series converge absolutely, converge conditionally or diverge.\\
\begin{tabbing}
\=xxxxxxxxxxxxxxxxxxxxxxxxxx\=xxxxxxxxxxxxxxxxxxxxxxxx\=\kill
\>(a) $\dsum_{n=1}^\infty \dfrac{\sqrt{n+1}-\sqrt{n}}{n^a}$
\>(b) $\dsum_{n=1}^\infty \left(1-\sqrt[n]{n}\right)^n$
\>(c) $\dsum_{n=1}^\infty \dfrac{(-1)^n\arctan(n)}{n}$\\[2ex]
\>(d) $\dsum_{n=1}^\infty (-1)^n \dfrac{n^{42}}{(n+1)!}$ 
\>(e) $\dsum_{n=1}^\infty \dfrac{1}{n^{1+1/n}}$ 
\>(f) $\dsum_{n=1}^\infty \Big(\dfrac{n}{n+1}\Big)^{n^2}$\\[2ex]
\>(g) $\dsum_{n=2}^\infty \dfrac{\cos n\theta}{\log n}$
\>(h) $\dsum_{n=3}^\infty \dfrac{(-1)^n}{\sqrt n (\log n)^2}$
\>(i) $\dsum_{n=1}^\infty  \dfrac{(-1)^n}{(2+(-1)^n)n}$ 
\end{tabbing}

\item Define an infinite product $\displaystyle \prod_{i=1}^\infty 1+a_i$ as  \vspace{-1ex}
$\displaystyle \lim_{n\to\infty} \prod_{i=1}^n 1+a_i = \lim_{n\to\infty} (1 + a_1)(1 + a_2) \cdots(1+a_n)$
when this limit exists.\vspace{-1ex}
\begin{parts}
\item Let $a_i \ge 0$. Prove that $\displaystyle \prod_{i=1}^\infty 1+a_i$ converges if and only if $\dsum_{i=1}^\infty a_i$ converges.
\\[-.7ex] \hint take logs.
\item Let $0 \le a_i < 1$. Prove that $\displaystyle \prod_{i=1}^\infty 1-a_i > 0$ if and only if $\dsum_{i=1}^\infty a_i$ converges.
\end{parts} 


\end{Ex}
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