Inverse of Power Series

Determining Coefficients of Inverse of Power Series

$$\begin{array}{rl}A(x)B(x)& =\sum _{n=0}^{\mathrm{\infty }}(\sum _{k=0}^{n}a(k)b(n-k))x^n\\ & =(a(0)b(0))x^0+(a(0)b(1)+a(1)b(0))x^1+(a(0)b(2)+a(1)b(1)+a(2)b(0))x^2+\dots \\ & =1.\end{array}$$

Inverse of Geometric Series

$$\begin{array}{rl}A(x)& =1+ax+(ax{)}^2+(ax{)}^3+\dots ,\\ B(x)& =1-ax.\end{array}$$

$$\begin{array}{rl}A(x)B(x)& =(1+ax+(ax{)}^2+(ax{)}^3+\dots )(1-ax)\\ & =(1+ax+(ax{)}^2+(ax{)}^3+\dots )-((ax)-(ax{)}^2-(ax{)}^3-\dots )=1.\end{array}$$