Suppose I have a function $f(x,\alpha)$ which has a convergent expansion in the small parameter $\alpha$:
$ f(x,\alpha)=\sum_{n=1\dots \infty,i=1\dots k} g(x,n,i)\alpha^n $
This expansion has $k$ different functions, $g(x,n,i)$ with contribute at any give order $n$.
Now suppose I compute an approximation only taking the first $m$ terms as:
$f_m(x,\alpha) = \sum_{n=1\dots m,i=1\dots10}g(x,n,i)\alpha^n$
Now suppose it is hard to compute the some of the $k$ functions for $g(x,m+1,i)$. Say these functions are for $i=l\dots k$. I will then try to improve my approximation of $f_m(x,\alpha)$ with the following new approximation:
$ f_{m}'(x,\alpha) = f_m(x,\alpha) + \sum_{i=1}^{l-1}g(x,m+1,i)\alpha^{m+1} $
Is this approximation necessary better? Is there any reason it might not be?
More precisely: Define the error: $\Delta_m = \left| f(x,\alpha)-f_m(x,\alpha)\right|$ and $\Delta_m' = \left| f(x,\alpha)-f_{m}'(x,\alpha)\right|$. Does $\Delta_{m+1}<\Delta_m$ imply $\Delta_{m}'<\Delta_m$.
I am asking this, because often in physics it is attempted to make the order of approximation consistent(i.e computing only approximations of the form $f_m$). I am now faced with a problem which for which there is a motivation to compute $f_m'$. I believe it will works well, but I don't want to be clear about the statements I make. I am also interested in the specific question in it's own right.