This is probably going to be a silly question, but here it goes: assume you have a complicated derivation where some function $f(x,\lambda)$ is manipulated in various ways (for example, into something like $\int_a^b dx \ |\partial_\lambda f(x,\lambda)|^2$ ).
Suppose one is interested in the case where $\lambda$ is small, and thus wants to Taylor-expand the final expression as $\lambda\to 0$. Does it ever matter when one performs such an expansion? Meaning, if one already starts with the expanded function, or expands it in one of the intermediate steps, should one always obtain the same result?
The result remains the same for the case where you $precisely$ replace a convergent power series by its sum function, or vice versa , $i.e.$ you don't neglect any of the higher order terms in the Taylor expansion for analytical approximations.
As an example, if $|2x| << 1$, $exp(2x)$ is approximately equal to $1 + 2x$, [neglecting higher order terms].
Also $exp(2x) = [exp(x)]^2 = (1 + x)^2 = 1 + 2x + x^2.$