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{\bf Problem.} (Science and Math Applications.)

If $f(x)=1+a↓1x+a↓2x↑2+a↓3x↑3+\cdots$, there is a unique power series
$g(x)=1+b↓1x+b↓2x↑2+b↓3x↑3+\cdots$ such that $f(x)\cdot g(x)=1$.
Write a program which, given $N≤20$ and $a↓1,a↓2,\ldots,a↓N$, computes
$b↓1,b↓2,\ldots,b↓n$. Apply it to these series, for appropriate values
of~$N$:
$$\displaylines{
\hfil f(x)=e↑x=1+x+{x↑2\over 2}+{x↑3\over 3!}+\cdots\hfil\cr
\hfil f(x)=\cos(x)=1-{x↑2\over 2!}+{x↑4\over 4!}-{x↑6\over 6!}+\cdots\hfil\cr
\hfil f(x)=\sqrt{1+x}=1+{1\over 2}x
-{1\over 2}\cdot {1\over 2}\,{x↑2\over 2!}+{1\over 2}\cdot {1\over 2}\cdot
{3\over 2}\,{x↑3\over 3!}-{1\over 2}\cdot {1\over 2}\cdot {3\over 2}\cdot
{5\over 2}\cdot {x↑4\over 4!}+\cdots\;.\hfil\cr}$$

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\noindent
{\bf Problem.} (Science and Math Applications.)

If $f(x)=a↓1x+a↓2x↑2+a↓3x↑3+\cdots$, then there is a unique power series
$g(y)=b↓1y+b↓2y↑2+b↓3y↑3+\cdots$ for which $g\bigl(f(x)\bigr)=x$
$\bigl($and also, $f\bigl(g(y)\bigr)=y\bigr)$. Write a program which, given
$N≤20$ and $a↓1,a↓2,\ldots,a↓N$, computes $b↓1,b↓2,\ldots,b↓N$. Apply
it to these series, for appropriate values of~$N$.
$$\displaylines{
\hfil f(x)=e↑x-1=x+{x↑2\over 2}+{x↑3\over 3!}+{x↑4\over 4!}+\cdots\hfil\cr
\hfil f(x)=\sin x=x-{x↑3\over 3!}+{x↑5\over 5!}-\cdots\hfil\cr
\hfil f(x)=3x+3x↑2+x↑3\hfil\cr
\hfil \hbox{(to get $g(y)=\root 3 \of{1+y}-1\,$)}\hfil\cr}$$


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\copyright 1984 Robert W. Floyd

First draft October 23, 1984 

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