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\begin{document}

\begin{center}
{\Large Warm-Up Problems and Lecture Problems

Solutions

April 18, 2003}
\end{center}

\begin{enumerate}

\item 
	\begin{enumerate}
	\item  Find the derivative of $\frac{1}{1-x}$.
		\\\sol  $\frac{d}{dx}\left(\frac{1}{1-x} \right) = \frac{1}{(1-x)^2}$.

	\item  Find a power series for $\frac{1}{(1-x)^2}$.
		\\\sol  Since we know the power series for $\frac{1}{1-x}$, we can take the derivative to get a power series for $\frac{1}{(1-x)^2}$.
		\begin{eqnarray*}
		\frac{1}{(1-x)^2} & = & \frac{d}{dx} \left( \frac{1}{1-x} \right) \\
		& = & \frac{d}{dx} \left(1+x+x^2+x^3+x^4+ \cdots \right) \\
		& = & 1+2x+3x^2+4x^3+\cdots \\
		& = & \sum_{n=0}^\infty (n+1)x^n = \sum_{n=1}^\infty nx^{n-1}
		\end{eqnarray*}

	\end{enumerate}

\inch

\item  Find a power series for $e^{2x}$.
	\\\sol  To find this power series, we can substitute $2x$ into the power series for $e^x$:
	\begin{eqnarray*}
	e^{2x} & = & \sum_{n=0}^\infty \frac{(2x)^n}{n!} = \sum_{n=0}^\infty \frac{2^nx^n}{n!} \\
	& = & 1+2x+\frac{4x^2}{2!}+\frac{8x^3}{3!}+\cdots
	\end{eqnarray*}

\newpage

\item  Find a power series, centered at $x=0$, for $\frac{1}{4+x}$
	\\\sol 
	\begin{eqnarray*}
	\frac{1}{4+x} &  = & \frac{1}{4(1+x/4)} = \left( \frac{1}{4} \right) \frac{1}{1+(x/4)} \\
	& = & \left( \frac{1}{4} \right) \sum_{n=0}^\infty \left( \frac{x}{4} \right)^n 
		= \left( \frac{1}{4} \right) \sum_{n=0}^\infty \frac{x^n}{4^n} 
		= \sum_{n=0}^\infty \frac{x^n}{4^{n+1}} \\
	& = & \frac{1}{4} + \frac{x}{4^2} + \frac{x^2}{4^3} + \frac{x^3}{4^4} + \cdots
	\end{eqnarray*}

\inch

\item  Find a power series, centered at $x=0$, for $\ln(4+x)$. 
	\\\sol  
	\begin{eqnarray*}
	\ln(4+x) & = & \int \frac{1}{4+x} \; dx \\
	& = & \int \left(\frac{1}{4}+\frac{x}{4^2}+\frac{x^2}{4^3}+\frac{x^3}{4^4}+\cdots\right) \; dx \\
	& = & C + \frac{x}{4}+\frac{x^2}{2\cdot 4^2}+\frac{x^3}{3\cdot 4^3}+\frac{x^4}{4\cdot 4^4}+\cdots
	\end{eqnarray*}
	Plug in $x=0$ to get $C=\ln 4$:
	\[
	\ln(4+x) = \ln(4) + \frac{x}{4}+\frac{x^2}{2\cdot 4^2}+\frac{x^3}{3\cdot 4^3}+\frac{x^4}{4\cdot 4^4}+\cdots
	\]

	

\newpage

\lecture

\item  Find the Taylor series for $\sin x$, centered at $x=\frac{\pi}{6}$.
	\\\sol  You can make a chart as in class or use the formula $c_n = \frac{f^{(n)}(a)}{n!}$.
	In any case, you should get:
	\begin{eqnarray*}
	\sin(x) & = & \frac{1}{2} + \frac{\sqrt{3}}{2}(x-\frac{\pi}{6})
	- \frac{1}{2}\frac{(x-\frac{\pi}{6})^2}{2!} 
	- \frac{\sqrt{3}}{2}\frac{(x-\frac{\pi}{6})^3}{3!}
	\\
	&&
	+ \frac{1}{2}\frac{(x-\frac{\pi}{6})^4}{4!} 
	+ \frac{\sqrt{3}}{2}\frac{(x-\frac{\pi}{5})^3}{5!}
	- \frac{1}{2}\frac{(x-\frac{\pi}{6})^6}{6!} 
	- \frac{\sqrt{3}}{2}\frac{(x-\frac{\pi}{6})^7}{7!}
	+ \cdots
	\end{eqnarray*}


\end{enumerate}


\end{document}