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Let $p(x)$ be a given polynomial in $x$, with $p(0)≠0$. Let $q(x)$ be the
power series for $1/p(x)$. Design a program to compute the first twenty
terms of~$q(x)$, given $p(x)$ of degree up to~20. Check it with these:
$$\eqalign{{1\over 1+x}&=1-x+x↑2-x↑3+x↑4\cdots\cr
\noalign{\smallskip}
{1\over 1-x-x↑2}&=1+x+2x↑2+3x↑3+5x↑4+8x↑5+13x↑6+\cdots\cr
\noalign{\smallskip}
{1\over 1+2x+x↑2}&={1\over (1+x)↑2}=1-2x+3x↑2-4x↑3+5x↑4-6x↑5+\cdots\cr
\noalign{\smallskip}
{1\over 1+x+{x↑2\over 2}+{x↑3\over 6}+{x↑4\over 24}\cdots}&=1-x+{x↑2\over 2}
-{x↑3\over 6}+{x↑4\over 24}+\cdots\cr}$$
and use it to develop the power series for $1/\cos x$, $x/\sin x$, and
$x/\ln(1+x)$.

Using the above program, converted into a procedure, develop a given power
series $r(x)=c↓0+c↓1x+c↓2x↑2+\cdots +c↓nx↑n$ into a continued fraction
$a↓0+{x\over a↓1+{x\over a↓2+{1\over\ddddots}}}$
using the relation
$r(x)=c↓0+{x\over r'(x)}$, where $r'(x)={1\over \left({r(x)-c↓0\over x}\right)}$.


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\line{\copyright 1985 Robert W. Floyd\hfill}
\line{First draft (not published) October 1, 1985;\hfill}
%revised: Date; subsequently revised.\hfill}

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