A(x)=a_{0}+a_{1}*x+a_{2}*x^{2}+...
B(x)=b_{0}+b_{1}*x+b_{2}*x^{2}+...
what conditions are needed to make A(x)B(x)=1 true for all real x
If the equation A(x)B(x)=1 is true for all real x, then the left side of the equation must be defined for all real x. So the series for A(x) and B(x) must both have infinite radius of convergence. Then the Cauchy product of the two series will also have infinite radius of convergence. The condition that this product should be equal to 1 for all x is
$\displaystyle \displaystyle a_0b_0=1,\quad\sum_{k=0}^na_kb_{n-k}=0$ for all $\displaystyle k\geqslant1$.