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{\bf Mathematics 112} \hfill {\bf Fall 1995}
\smallskip
\centerline{\bf Projects-Based Sections (Barzilai and Henderson)}
\dba
\centerline{\bf Taylor-Series Cookbook}
\bs

{\bf Recipes:}

(i) {\it From scratch:}

Directly compute $a_n = {f^{(n)}(c)\over{n!}}$ to obtain the Taylor
series for f(x) at c, $\sum_{n=0}^{\infty} a_n (x-c)^n$

\ms
(ii) {\it Substitution:}

Write your function $f(x)$ as $g(2x)$ or $g(\sqrt{x})$ or $g(h(x))$
for some $g(x)$ whose Taylor series you know, to get the Taylor series
for $f(x)$

\ms
(iii) {\it Differentiation or Integration:}

If your function $f(x)$ is equal to ${d\over dx}(g(x))$ for some
$g(x)$ whose Taylor series you know as $\sum_{n=0}^{\infty} a_n
(x-c)^n$, then you have a Taylor series for $f(x)$, namely,
$\sum_{n=0}^{\infty} n a_n (x-c)^n-1$. Similarly, if your $f(x)$ is
equal to the integral of some $g(x)$ whose Taylor series you know, you
can integrate $g$'s Taylor series term-by-term to get $f$'s Taylor
series. {\it Note:} In this case you would also have to solve for C,
the constant of integration.

\ms
(iv) {\it Creative Combos:}

If the Taylor series for $f(x)$ and $g(x)$ respectively are 
$\sum_{n=0}^{\infty} a_n (x-c)^n$
and $\sum_{n=0}^{\infty} b_n (x-c)^n$, what is the Taylor series for
$h(x)$ if $h(x) = f(x) + g(x)$? if $h(x)=x f (x)$?

\bs
\centerline{$\star\star\star\ \ \ \ \ \ \ \star\star\star\ \ \ \ \ \ \ \star\star\star$}

\bs
\ss
Cook up a storm by finding the Taylor series for the the following
cornucopia of functions:

\ms

${1\over{x+1}}$  \hfill ${ln(x+1)}$  \hfill ${1\over{(x+1)^2}}$ 
\hfill ${{x}\over{1-x}}$ 

\ms

${{1}\over{1+{4x^2}}}$  \hfill ${tan^{-1}(2x)}$ \hfill
${{1}\over{4+{x^2}}}$ \hfill ${{1}\over{1+{x^2}}}$

\ms

How about $ln({{1+x}\over{1-x}})$ (Try re-writing
this, first) and ${2\over{3x+4}}$?

\bs
\centerline{$\star\star\star\ \ \ \ \ \ \ \star\star\star\ \ \ \ \ \ \ \star\star\star$}
\bs

For the finishing touch, compute the Radii and Intervals of
convergence of these Taylor series.

\ms
(C) Copyleft 1995 by Harel Barzilai. ``Share and Enjoy.'' Permission
given for nonprofit educational use.

\dba
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\bye                      
