Calculating Coefficients of Infinite Power Series

[zetafunction]

given the infinite power series

$$f(x)= \sum_{n=0}^{\infty}a_ {n}x^{n}$$

if we know ALL the a(n) is there a straight formula to get the coefficients of the b(n)

$$\frac{1}{f(x)}= \sum_{n=0}^{\infty}b_ {n}x^{n}$$

for example from the chain rule for 1/x and f(x) could be obtain some combinatorial argument to get the b(n) from the a(n) ??

 

[mathman]

Not usually. f(x)=x. There is no power series for 1/f(x).

 

[g_edgar]

Take the case a_0 not 0. We can obtain the b_n by *long division* of power series.