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!}<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.

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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