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

Mathematics Asked on January 3, 2022

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
$$sum_{k=0}^infty sup_{xin[-M,M]}Big|(-1)^kfrac{[ln(1+x^2)]^k}{k!}Big|=sum_{k=0}^infty sup_{xin[-M,M]}frac{[ln(1+x^2)]^k}{k!}=sum_{k=0}^inftyfrac{[ln(1+M^2)]^k}{k!}
I thought I could use a reductio ad absurdum argument to prove it does not converge uniformly on $$mathbb{R}$$, but I didn’t manage to make it work.

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 $$sup_{x in mathbb{R}} |f_k(x)| = infty$$.

Answered by RRL on January 3, 2022

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