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

Question

I would like to prove that the series of functions $$ \sum_{k=0}^{\infty}(-1)^k\frac{[\ln(1+x^2)]^k}{k!} $$ 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 $$ \sum_{k=0}^\infty \sup_{x\in[-M,M]}\Big|(-1)^k\frac{[\ln(1+x^2)]^k}{k!}\Big|=\sum_{k=0}^\infty \sup_{x\in[-M,M]}\frac{[\ln(1+x^2)]^k}{k!}=\sum_{k=0}^\infty\frac{[\ln(1+M^2)]^k}{k!}<\infty. $$ 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.

Answer 1

For $\sum_{k} f_k(x)$ to converge uniformly for all $x \in \mathbb{R}$ it is necessary that $|f_k(x)| \to 0$ uniformly, and, equivalently, $\sup_{x \in \mathbb{R}} |f_k(x)| \to 0$ as $k \to \infty$.

In this case $\sup_{x \in \mathbb{R}} |f_k(x)| = \infty$.