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\title{MATH3210 - Spring 2024 - Section 004}
\author{Homework 10}

\begin{document}

\maketitle

\begin{problem}
Prove the divergence criteria of the comparison test. That is, show that if $\set{a_n}$ and $\set{b_n}$ are sequences such that $a_n \ge b_n \ge 0$ for every $n$, and $\sum_{n=1}^\infty b_n$ diverges, then $\sum_{n=1}^\infty a_n$ diverges. 

Use the following scheme: Let $t_m = \sum_{n=1}^m b_n$ be the sequence of partial sums for $b_n$. Show that if $\sum_{n=1}^\infty b_n$ diverges, then $t_m$ must diverge to infinity. Then show that the sequence $t_m$ diverges to $\infty$, then so does $s_m = \sum_{n=1}^m a_n$.
\end{problem}

\begin{problem}
Determine whether the series converges or diverges. Prove that your answer is correct using the following tools only: the term test, the comparison test, the ratio test, and the integral test.

\begin{enumerate}[(a)]
\item $\displaystyle\sum_{n=1}^\infty \cos(n)$
\item $\displaystyle\sum_{n=1}^\infty \dfrac{1}{n^2+1}$
\item $\displaystyle\sum_{n=1}^\infty \dfrac{n^8}{n!}$
\item $\displaystyle\sum_{n=1}^\infty \dfrac{n^3 + 3^n}{5^n}$
\end{enumerate}
\end{problem}

\begin{problem}
Let $f_n : [a,b] \to \R$ be a sequence of positive functions such that $\sum_{n=1}^\infty f_n(0)$ converges, and each $f_n$ is $L_n$-Lipschitz. Show that if $L = \sum_{n=1}^\infty L_n$ converges, then $\sum_{n=1}^\infty f_n$ converges uniformly to an $L$-Lipschitz function.
\end{problem}

\begin{problem}
Let $f(x) = xe^x$, Find a number $N$ such that the Taylor approximation of order $N$ is within $.1$ of $f(x)$ on the interval $[0,1]$. [{\it Hint}: First, find and prove a formula for $f^{(k)}(x)$ by induciton, then bound $f^{(k)}(x)$ on $[0,1]$ using this formula by a number depending on $k$]
\end{problem}

\end{document}


