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%  Homework for the course "Calculus II", 
%  Spring semester, 2024, Wei-Cheng Wang.

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\begin{document}

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\noindent
{\scriptsize Calculus II, Spring 2024 
(http://www.math.nthu.edu.tw/\~\,wangwc/)} \hfill 

\begin{center}
\large
Homework 04   
\normalsize
\end{center}

\vspace{.3in}

%           The questions!

\begin{enumerate}

% \item % The Ratio and Root Tests
% Section 10.5: (optional, to answer a question raised in class)
% Take the following Stirling's formula
% for granted:
% \begin{equation} \label{Stirling}
 % \lim_{n\to\infty} {e^n n! \over n^n \sqrt{n} } = \sqrt{2\pi}
% \end{equation}
% You can easily find some readable proof via Google search if you are curious.
% Use \eqref{Stirling} to determine whether
% $\displaystyle{ \sum_{n=1}^\infty { (2n)! \over 4^n n! n! }}$
% converges or not.
% 
\item % The Integral Test. bounds for the remainder 
Section 10.3: Problem 51.

Hint: use "Bounds for the Remainder in the Integral Test" on page 611.


\item % Alternating Series and Conditional Convergence. 
Section 10.6: Problems 11, 25, 26, 28, 29, 30, 35, 39, 41, 49, 53.

Remark: 
Take the following for granted:
The sequences
$a_n = \displaystyle{\ln n \over n}$,
$b_n = \displaystyle{\tan^{-1} n \over n^2+1}$,
$c_n = \displaystyle{\ln n \over n - \ln n}$,
$d_n = (\sqrt{n+1}-\sqrt{n})$
are all decreasing for $n>3$. 
% You can try to verify this fact,
% or take it for granted if not successful.

\item %  Power series
Section 10.7: problems 7, 11, 15, 19, 23, 29, 40, 43, 47. 

Hint for problem 40: 
$\displaystyle{n \over n+1}
 = \Big( 1 + {1\over n} \Big)^{-1}$.

%  Examples of convergence on  ( ), ( ], [ ) and []
\item %  Power series
Section 10.7:
Find a power series that converges on
$(1,3)$ and diverges elsewhere.
Do the same for
$(1,3]$,
$[1,3)$ and
$[1,3]$, respectively.

% \item %  Power series
% Section 10.7:
% Find the first few terms of the power series representation of
% $$
% {1 -x^2 + x^4 -\cdots \over 1 - {x^2 \over 2!} + {x^4 \over 4!} - \cdots}.
% $$

% \item %  Power series
% Section 10.7: 51, 55, 57, 60. 
% 
% \item %  Power series
% Section 10.7: Use the power series representation of ${1 \over 1-x}$
% to find the power series representation of $\ln(1-x)$ on $|x|<1$.
\end{enumerate}

\end{document}
