Reversion of power series So, I just heard about this method. How does one determine the coefficients, and what is it used for? For example, given $$ y = x - \frac{x^3}{6} + \frac{x^5}{120} + O(x^7)$$ reversion would give a series for $x$ in terms of $y$, correct? And then this expansion could be used to expand $y$ in ascending powers of itself?

I think how you go about inverting the series is to first do this:

$$x = y + \frac{x^3}{6} - \frac{x^5}{120} - O(x^7),$$

so that

$$x = y + \frac{1}{6}(y + \frac{x^3}{6} - \frac{x^5}{120} - O(x^7))^3 - \frac{1}{120}(y + \frac{x^3}{6} - \frac{x^5}{120} - O(x^7))^5 + O(y^7)$$

Then gather the terms of order less than $y^7$ and discard those $y^7$ and higher. If you still have terms with powers of $x$ that are multiplied by powers of $y$ with order less than $y^7$, substitute the first expression again until you don't.