I encountered this problem from quantum field theory. Let $f(x)$ and $g(x)$ be two smooth functions. Now I want to find the taylor expansion of the function $$f(x+g(x)).$$
The direct answer is to treat it as a composite function and then use the chain rule and plug it into the formula of the Taylor series. However, I am wondering if the following series makes sense.
$$f(x+g(x))=f(x)+f^{\prime}(x)g(x)+\frac{1}{2}f^{\prime\prime}(x)g(x)^{2}+\cdots ?$$
Best Answer
Promoting from and elaborating on a comment, because it seems to have resolved the OP's confusion:
It would make sense if you wrote $$ f(x_0+g(x))=f(x_0)+f^{\prime}(x_0)g(x)+\frac{1}{2}f^{\prime\prime}(x_0)g(x)^{2}+\cdots $$ and $g(x)$ was smaller than the radius of convergence of $f$ when expanded about $x_0$. You could even then substitute in an expansion of $g$ about $x_0$, something like $$ g(x) = g(x_0) + g'(x_0) \Delta x + \frac12 g''(x_0) \Delta x^2 + \cdots $$ with $\Delta x \equiv x - x_0$. But you do have to pick a "base point" for your Taylor expansion; the coefficients of the series are always the derivatives of the function evaluated at some base point, while the "powers" are some variable corresponding to your deviation from that base point.