Recall the Fourier-Legendre expansion of any given function $f$ of $L^2(-1,1)$:
$$\frac{1}{2}\sum\limits_{n=0}^{N}\left[(2n+1)\int_{-1}^{1}f(s)P_n(s)ds\right]P_n\underset{N\to\infty}{\longrightarrow}f$$
Where $P_n$ denotes the Legendre Polynomials, and "$\underset{N\to\infty}{\longrightarrow}$" means convergence in $L^2(-1,1)$, i.e the $||\cdot||_2$ norm of the difference converges towards $0$.
Now, convergence in $L^2(-1,1)$ doens't imply pointwise convergence, but I have found the following result:
$$\text{If }\sum\left((2n+1)\int_{-1}^{1}f(s)P_n(s)ds\right)_{n\in\mathbb{N}}\text{ converges absolutely,}$$
$$\text{then the aforementioned Fourier-Legendre expansion of }f\text{ converges pointwise towards }f.$$
The thing is, after some numerical tests it looks to me that there exists absolutely no function $f$ that checks this condition: the series always seems not only to not converge absolutely, but even to diverge dramaticaly.
Would someone be able to provide me with an example of a function $f$ that respects this condition?