I am interested in the sum over composite functions, and would like to approximate the sum by the sum of the Taylor expansions (to first order to make it simple). Can I make a statement about the error for the sum? Would it be equal to the sum of the errors of approximations? Would the errors potentially average out? Here the maths.

I linearly expand the composite functions (index $n$ labels the functions) $f_n(g(x))$ around $x_0$ as

$$ f_n(g(x)) approx f_n(g(x_0)) +f’_n(g(x_0))g'(x_0)(x-x_0) + o((x-x_0)^2) $$

I am interested in the sum $sum_n f_n(g(x))$. By linearity of the derivative, the sum is approximated as

$$ sum_n f_n(g(x)) approx sum_n f_n(g(x_0)) +f’_n(g(x_0))g'(x_0)(x-x_0) + o((x-x_0)^2)$$

Now, what statement can be made about the error?

- It is proportional to $(x-x_0)^2$ ?
- Can the error for each function $n$ be estimated as $$frac{d^2/dx^2 f_n(g(x))}{d/dx f_n(g(x))}|_{x=x_0} = f_n”(g(x))|_{x=x_0} + frac{g”(x)}{g'(x)}|_{x=x_0} $$
- If the above can be used, how would the error on the sum behave? Can something be said? Do the errors simply sum up or do they average out?