I am working through the first chapter of Dunham's Calculus Gallery (available here), and I am trying to see how Newton derived the infinite series for the exponential function, knowing a series for the natural logarithm:
$z = \ln (1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} \dots$
Dunham says Newton starts with a series $z = f(x)$ and obtains $x = f(z)$ by inverting it (Wolfram calls this reversing a series).
The method is to guess that $x \approx z$, so $x = z + p$, where $p$ is an error term. $p$ can be evaluated by plugging into $f(x)$, and keeping only the simplest terms. He gives two examples including deriving the sine from the inverse sine. I have worked through these as well.
For this problem (the exponential), two cycles gives $x = z + \frac{z^2}{2} + \frac{z^3}{3!} + \dots$ which suggests I'm on the right track.
My two (related?) problems are: (i) how to deal with the fact that the original series was for $\ln (1 + x)$ and (ii) I am missing the leading factor of 1 in the exponential series.
A change of variable turns into a mess. I think I'm close but I'm not sure how to clean this up.