Inverting Series: Solving for z in Powers of x

[qbslug]

How do you invert a series? Can't find any information on it.
I need to invert a function and solve for z in powers of x

$$x = \sum_{n=1}^\infty\\b_n z^n$$

I'm guessing we can just say

$$z = \sum_{n=1}^\infty\\a_n x^n$$

But if this is right I don't know why its right or if it can be proven.
I would like to know the general idea behind inverting series functions and why it works.

 

[jackmell]

Here's a link:

http://mathworld.wolfram.com/SeriesReversion.html

 

[HallsofIvy]

Pretty darn complicated! Note that the term "reversion" is used since "inversion" could be confused with finding the reciprocal, 1/z.

 

[qbslug]

Ok thanks I guess I wasn't getting much information about it because I was referring to it as "inversion"