My aim is to find an example where the CLT is true but not the following (equivalent to Lindeberg's) condition:
Find a sequence of independent $(X_k)\sim\mathcal{N}\left(0,\sigma^2_k\right)$, so that they respect the central limit theorem but:
$\displaystyle\lim_{n \to +\infty}\frac{\max_{1\leqslant k\leqslant n}\sigma^2_k}{\sum_1^n\sigma^2_k}\ne0$
My first try is to take $\sigma^2_k=\displaystyle\frac{1}{k^2}$ so that
$\max_k\sigma^2_k=1$ and $\displaystyle s^2_n=\sum_1^n\frac{1}{k^2}\to L<+\infty$.
But what if I had $s^2_n\to\infty$?