Finite truncation of an expansion

The truncation through order $N$ of $\sum_{j\ge0}a_j\phi_j(\varepsilon)$ is the finite sum

S_N(\varepsilon)=\sum_{j=0}^N a_j\phi_j(\varepsilon).

If an actual function $f$ is specified, its remainder is $R_N=f-S_N$ . An asymptotic scale orders the retained contributions by size.

Meaning of an error bound

A full asymptotic expansion gives $R_N=o(|\phi_N|)$ and, by using the next coefficient, $R_N=O(|\phi_{N+1}|)$ . Constants can depend on $N$ . A formal truncation by itself provides no bound on a remainder of an actual function.