I want to explain the following limit: $$ \sum_{k=1}^n\binom{n}{k}a_k \stackrel{n\ll1\\}{\,\,\,=\,\,\,}n\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}a_k+o(n) $$ Is there a simple way to prove it? The unusual thing is that $n$ is considered a natural number at the beginning, to properly define the sum from 1 to $n$, while to do the limit a sort of analytical continuation of $n$ to real numbers is needed.
(I'm studying the following paper: https://hal.science/jpa-00232897/document , the authors use the previous limit between Eqs. (15) and (16). The continuation for $n$ from $\mathbb{N}$ to $\mathbb{R}$ was shown to be correct in some applications to statistical mechanics of disordered systems. )
Edit
As pointed out in a comment, it is possible to perform the $n\rightarrow0$ limit for the so-called "generalized binomial coefficients": $$ \lim_{n\rightarrow0}\frac{1}{n}\binom{n}{k}=\frac{(-1)^{k-1}}{k} $$ The result of the whole limit in the first equation is therefore reasonable, but a question still holds: why the sum from 1 to $n$ become a sum from 1 to $\infty$ as $n\rightarrow0$?