Uniform convergence of $sum_{k=0}^{infty}(-1)^kfrac{[ln(1+x^2)]^k}{k!}$ on $mathbb{R}$.

I would like to prove that the series of functions
$$
sum_{k=0}^{infty}(-1)^kfrac{[ln(1+x^2)]^k}{k!}
$$
does not converge uniformly on $mathbb{R}$. I proved that it does converge totally (and thus uniformly, and pointwise) on all the compact sets $[-M, M]$. The total convergence on the compact intervals $[-M,M]$ is quite straightforward since
$$
sum_{k=0}^infty sup_{xin[-M,M]}Big|(-1)^kfrac{[ln(1+x^2)]^k}{k!}Big|=sum_{k=0}^infty sup_{xin[-M,M]}frac{[ln(1+x^2)]^k}{k!}=sum_{k=0}^inftyfrac{[ln(1+M^2)]^k}{k!}<infty.
$$
I thought I could use a reductio ad absurdum argument to prove it does not converge uniformly on $mathbb{R}$, but I didn’t manage to make it work.