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\newcommand{\myCourseNumber}{Math 142A}
\newcommand{\myAssignmentDueDate}{Friday, November 4, 2022}
\newcommand{\myAssignmentTitle}{Homework Assignment 4}
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\begin{document}

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\begin{questions}
    
\question Let $(s_n)$ and $(t_n)$ be bounded sequences of nonnegative real numbers.  Prove that
\[ \lim\sup s_n \, t_n \leq \left( \lim\sup s_n \right) \left( \lim\sup t_n \right). \]

\question Prove that $(s_n)$ is bounded if and only if $\lim\sup |s_n| \in \mathbb{R}$ (that is, $\lim\sup |s_n| < +\infty$).

\question Let $(s_n)$ be a bounded sequence of nonzero real numbers.  Prove that
\[ \lim\inf \left| \frac{s_{n+1}}{s_n} \right|  \leq \lim\inf \left| s_n \right|^{1/n} . \]

\question Let $(s_n)$ be a sequence of nonnegative numbers.  For each $n$, define $\displaystyle \sigma_n = \frac{1}{n} \left(s_1 + s_2 + \cdots + s_n \right)$.  [Recall from Homework 2 that $(\sigma_n)$ is called the sequence of Ces\`{a}ro means for $(s_n)$.]
\begin{parts}
\part Show that \enspace $\displaystyle \lim\inf s_n \leq \lim\inf \sigma_n \leq \lim\sup \sigma_n \leq \lim\sup s_n$.
\part Show that if \ $\lim s_n$ \ exists, then \ $\lim \sigma_n$ \ exists and \ $\lim \sigma_n = \lim s_n$.
\part Exhibit an example for which \ $\lim \sigma_n$ \ exists, but \ $\lim s_n$ \ does not exist.
\end{parts}

\question Show that if \ $\sum a_n$ \ and \ $\sum b_n$ \ are convergent series of nonnegative numbers, then \ $\sum \sqrt{a_n \, b_n}$ \ converges.

\question Find a series \ $\sum a_n$ \ which diverges by the Root Test but for which the Ratio Test gives no information.

\question Let $(a_n)$ be a sequence of nonzero real numbers such that the sequence \ $\left(\frac{a_{n+1}}{a_n}\right)$ \ is a constant sequence.  Show that \ $\sum a_n$ \ is a geometric series.

\question Let \ $(a_n)_{n \in \mathbb{N}}$ \ be a sequence such that \ $\lim\inf |a_n| = 0$.  Prove there is a subsequence \ $\left( a_{n_k} \right)_{k \in \mathbb{N}}$ \ such that \ $\displaystyle \sum_{k=1}^{\infty} a_{n_k}$ \ converges.

\question \begin{parts}
\part Exhibit an example of a divergent series \ $\sum a_n$ \ for which \ $\sum a_n^2$ \ converges.
\part Show that if \ $\sum a_n$ \ is a convergent series of nonnegative terms, then \ $\sum a_n^2$ \ also converges.
\part Exhibit an example of a convergent series \ $\sum a_n$ \ for which \ $\sum a_n^2$ \ diverges.
\end{parts}

\question Prove that if $(a_n)$ is a decreasing sequence of positive real numbers and if \ $\sum a_n$ \ converges, then \ $\displaystyle \lim_{n \to \infty} n a_n = 0$

\end{questions}

\end{document}
