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

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\def\V{\mathcal V}
\def\L{\mathcal L}
\def\S{\mathcal S}
\def\T{\mathcal T}
\def\U{\mathcal U}
\def\A{\boldmath A}
\def\M{\boldmath M}
\def\l{\mbox{\boldmath $l$}}
\def\r{\mbox{\boldmath $r$}}
\def\v{\mbox{\boldmath $v$}}
\def\ds{\displaystyle}
\def\limi{\displaystyle{\lim_{n\to \infty}}}
\def\sumi{\displaystyle{\sum_{n=1}^\infty}}
\def\sumzi{\displaystyle{\sum_{n=0}^\infty}}
\def\sumk{\displaystyle{\sum_{n=1}^k}}
\def\sumkzi{\displaystyle{\sum_{k=0}^\infty}}
\def\limk{\displaystyle{\lim_{k\to \infty}}}


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\begin{document}

%          The title and header.

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

\begin{center}
\large
Homework 05   
\normalsize
\end{center}

\vspace{.3in}

%           The questions!

\begin{enumerate}

\item %  Power series
Section 10.7: 51, 55, 57, 60. 

\item %  Power series
Section 10.7: Use the power series representation of $\ds{1 \over 1 \pm x}$
to find the power series representation of $\ln(1 \pm x)$ on $|x|<1$.

\item %  Power series
Section 10.7:
% Assuming the
% power series representation of
Find the first three nonzero terms of the power series representation of 
$$
\ds{1 -x^2 + x^4 -\cdots \over 1 - \ds{x^2 \over 2!} + \ds{x^4 \over 4!} - \cdots}
$$
by "long division".

\item % 10.8: Taylor and MacLaurin Series.
Section 10.8: Problems 5 ($n=3$), 7 ($n=3$), 15, 23, 29, 35.

% Remark for problems 5, 7:
% Taylor polynomial of order $n$ generated by $f$ at $a$ is simply the first $n+1$
% terms of the Taylor series generated by $f$ at $a$.
% See the definition on page 642.
% Do problem 5 and 7 for $n=3$.

Hint for problem 5: 

Method 1: Find $f^{(n)}(a)$ by repeated differentiation.

Method 2: 
% another way of finding the answer (without performing all the derivatives) is to 
Write 
$\ds{1\over x} = \ds{ 1 \over x-2+2} 
 = \ds{1\over 2}\ds{ 1 \over \left(1 + \displaystyle{x-2 \over 2} \right)} $ 
and use the formula for geometric series
to find the power series representation of $f(x)$. This will give you 
Taylor series generated by $f$ at $a$ (Theorem A, Lecture 09)
and therefore Taylor polynomial generated by $f$ at $a$.

Hint for problem 35: 

Method 1: Find $f^{(n)}(a)$ by repeated differentiation.

Method 2 (easier):
Use problem 2 above, Theorem 19 of section 10.7 and finally Theorem A, Lecture 09.

Update: also need to use a result from Section 10.9, 
Example 2: $\sin x = x - \ds{x^3 \over 3!} + \ds{x^5 \over 5!}- ...$ valid for 
all $x \in \mathbb{R}$ (that is, radius of convergence $R>0$).

\end{enumerate}

\noindent{\bf Theorem A} (See page 10 of Lecture 09):

\noindent If $f(x)$ has a power series representation on $|x-a|<R$ with \underline{$R>0$},
then $T_{f,a}(x) = f(x)$.

In other words:

\noindent If $f(x) = \ds{\sumkzi a_k (x-a)^k}$ on $|x-a|<R$ with \underline{$R>0$},
then $T_{f,a}(x) = \ds{\sumkzi a_k (x-a)^k}$.
\end{document}
