Question

5. A function f has Taylor series (at 0) f(x)=0+2x+ 4x2/2! + 3x3/3!+...

Assume f−1 exists. Find as much of the Taylor series of f−1 (at 0) as you can. (Since you only know the first few terms of the Taylor series for f, you can only figure out f−1.

(Hint: There are two ways of doing this problem. One is get the derivatives of f−1 from knowing the derivatives of f; we talked about the first derivative in January and higher derivatives can be found by implicit differentiation with the chain rule and the equation f−1(f(x)) = x. Another is to use the equation f−1(f(x)) = x, assume f−1 has a power series expansion with unknown coefficients cn, plug in for f(x), and try to solve for the cn. If you want to try a web search, this is called “Lagrange inversion,” but I’m not expecting you to understand any of the sources that come up.)