Example

Let $g(x) = x^\lambda = \exp(\lambda \log x)$, of order $+\infty$ relative to $e^\lambda$. Taking $p = 2$ one has

whence (V, p. 225)

and similarly

consequently

One proceeds similarly (for $r_n$) when $g$ is of order $-\infty$ relative to $e^\lambda$.

Now if $g$ is of finite order $\mu$ relative to $e^x$, and if, for example, $\sum_{n=1}^\infty g(n) = +\infty$, one can write

where $f_1(n) = g(n) - \frac{\mu}{1 - e^{-\mu}} \int_1^n g(t)\, dt \ll g(n)$ by prop. 6 of V, p. 239. If one has a principal part $c g_1(n)$ of $f_1(n)$ relative to $\mathscr{E}'$, and if one can again apply prop. 6 to the function $g_1$ one will obtain a primitive equivalent to $\sum_{m=1}^n f_1(m)$ if $\sum_{n=1}^\infty g_1(n) = +\infty$, and equivalent to $\sum_{m=n+1}^\infty f_1(m)$ in the opposite case (in the latter case one writes $\sum_{m=1}^n f_1(m) = C - \sum_{m=n+1}^\infty f_1(m)$, with $C = \sum_{n=1}^\infty f_1(n)$).

Step by step one can thus eventually obtain an expression for $s_n$ as the sum of a certain number of primitives each of which is negligible with respect to the previous, of a term remaining negligible relative to the last primitive written, and finally a constant (the case where the remainder term tends to 0). It then remains to expand each of the primitives obtained with respect to $\mathscr{E}'$ (cf. V, p. 235).