Can we find reals $x,y$ so $ \lim\limits_{n\to \infty} (a_n-x n)=y$? I used the following lemma for $a_n=\sum\limits_{k=1}^n \frac{k}{n} e^{\frac{k}{n^2}}$ from "Exercices de mathématiques: oraux X-ENS (Analyse I)", by Francinou, Gianella, and Nicolas (2014, exercise 4.26):
Let $f\colon\mathbb{R}\to\mathbb{R}$ be differentiable at $0$, and such that $f(0)=0$. Letting $s_n\stackrel{\rm def}{=} \sum\limits_{k=0}^n f\!\left(\frac{k}{n^2}\right)$ for $n\geq 1$, the limit of the sequence $(s_n)_{n\geq 1}$ is $\frac{f'(0)}{2}$.
The answer is no: set $a_n = 1/2 + 1/n$ for $n$ even and $a_n = 1/2 + e^{-n}$ for $n$ odd.
PS: très saines lectures, bravo.