Knowing the expansion of a function, how can we find its expansion using the inverse of x? [migrated]

If we have a function like:

$$\text{f[x \_ ]:=}\sum _{i=0}^{\infty } a_ix^i$$

where we can find / know the $$a_i$$ coefficients, but not really for which function it will converge.

How can we find $$f[x]$$ but using the inverse of $$x$$ instead? Something like this?

$$\text{f[x \_ ]:=}\sum _{i=0}^{\infty } \frac{b_i}{x^i}$$

The main problem is that the first form of $$f[x]$$ does not converge properly for positive values greater than one, since it comes from a Taylor series.