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.