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{{\bf Calculus Lab II, Spring 2004}\\
{\bf Pre-Lab 1:  Review of Power Series}}
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Next week I'll give you a short lab based on power series.  This
week, I would like for you to review the basics on power series
and Taylor series.  Include your typewritten (or graphical) answers
to the following questions in the lab writeup.

\begin{enumerate}
\item A {\it power series based at $x=a$} is a function of the
form:
$$
P(x)=\sum_{n=0}^{\infty}c_n(x-a)^n
$$
where $c_n$ is a constant, for all $n$. What does it mean to take
an ``infinite'' sum?
\item What is the radius of convergence for a power series?  What
is the ``Ratio Test''?
\item Can you differentiate/integrate a power series?  What is the
result after doing this to $P(x)$ defined in Problem 1?
\item What is a Taylor series?  How is it different than a
Maclaurin series?
\item What is the meaning of ``$n^{\rm th}$ Taylor polynomial''?
What is the meaning of $R_n(x)$ (used in Stewart's Calc text)?
\item How does using a Taylor series allow us to integrate
functions we otherwise could not?
\item Do {\it all} functions have a power series representation?
How can you tell if it does?
\item It is shown in Calculus that, if $|x|<1$, then
\begin{equation}
\label{Eq1}
\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n
\end{equation}
Show graphically that the partial sums approximate $1/(1-x)$ by
plotting some.  That is, define:
$$
f_0(x)=\frac{1}{1-x}, \quad f_3(x)=1+x+x^2+x^3,
$$
$$
f_6(x)=1+x+x^2+x^3+x^4+x^5+x^6, \mbox{ etc. }
$$
and plot $f_0$ along with functions of increasing degree. Careful
in defining the domain and range windows!
\item This refers to the previous problem:  Integrate both sides
of Equation \ref{Eq1} and repeat the graphs.
\end{enumerate}
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{\bf Power Series and Maple}
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In the example below, we use Maple and the Taylor series to
approximate the antiderivative of ${\rm e}^{x^3}$
\begin{verbatim}
F:=int(exp(x^3),x);
\end{verbatim}
Your answer has $\Gamma(x)$ in it- this is the Gamma function,
which is itself defined as an integral.  To satisfy your
curiousity, it is defined as:
$$
\Gamma(n)=\int_0^{\infty}x^{n-1}{\rm e}^{-x}\,dx
$$
If $n$ is a positive integer, then
$$
\Gamma(n)=(n-1)!
$$
The $\Gamma$ function is therefore a generalization of the
factorial for non-integers (and it is used extensively in
statistics).

Getting back to the problem, we'd like an approximation to the
antiderivative of ${\rm e}^{x^3}$.  Given Maple's output, we
continue:
\begin{verbatim}
Ft:=taylor(F,x=0,12);
     x+1/4*x^4+1/14*x^7+1/60*x^10+O(x^13)
\end{verbatim}
Maple uses the ``Big-Oh'' notation to tell us that the next term up
will be $x^{13}$.  There is a formal definition for the ``Big-Oh'', but
an intuitive idea is sufficient for now (we define it formally in Numerical Analysis).
We now need to convert this power series into a regular polynomial so that
we can plot it, differentiate it, integrate it, etc.  This is done as:
\begin{verbatim}
Fp:=convert(Ft,polynom);
\end{verbatim}
Is this a very good approximation?  One way to check is to
compare the derivative of this to the series for ${\rm e}^{x^3}$.
That is, compare the Maple output for:
\begin{verbatim}
diff(Fp,x);
taylor(exp(x^3),x,11);
\end{verbatim}



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