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

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

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Homework 02   
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\vspace{.3in}

%           The questions!

\begin{enumerate}

% Section 8.8: % Improper integrals
\item
Section 8.8: 
Problems: 35, 39, 41, 45, 55, 65, 66.%, 77, 81, 83.
% Problems: 7, 13, 19, 21, 25, 31, 35, 39, 42, 45, 46, 55, 65, 66, 77, 81, 83.
% problem 77 is harder,need more hint
% Hint for problem 77:

% Section 8.8: % Improper integrals
\item
Section 8.8: 

Let $f(x) = \ds{1 \over x^{0.5} \ln x}$.
Is the improper integral $\ds{\int_{e}^\infty f(x) \ dx}$ convergent?

Hint: Try not to find the anti-derivative of $f(x)$, but instead find a suitable $g(x)$
for comparison using Theorem 2 or Theorem 3.
See also Example 5-7 of Lecture 02 on how to find $g(x)$.

% Section 8.8: % Improper integrals
\item
Section 8.8: 

Do the same for $\ds{\int_0^{1\over e} \ds{1 \over x^2 |\ln x|} \ dx}$.

% Section 10.1: Sequences
\item 
Section 10.1: % Sequences
  Problems 46, 53, 59, 63, 67, 69, 73, 81, 87, 89.
% 14e: Section 9.1
% Problems 50, 57, 63, 67, 71, 79, 83, 91, 97, 135. 

Hint for problem 73: Read Appendix A.5, "Limit 6". %83

Hint for problem 87: Multiply by $\displaystyle{ n+\sqrt{n^2-n} \over n+\sqrt{n^2-n} }$. %97

Hint for problem 89: Use Theorem 4 and Fundamental Theorem of Calculus + L'H\^{o}pital's Rule. 
% Hint for problem 125: See page 2 and page 3 in v02 of lecture 02 note to see the revised
% (simpler) argument on how to find $N$. %135

% Section 10.2:  Infinite series
\item % Infinite series
Section 10.2:  Problems 31, 33, 43, 61, 63, 65, 71, 78.

\end{enumerate}

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