Consider $X = (-1)^{k}k$ and $\mathbb{P}(X = (-1)^{k}k) = \frac{const}{k^{2}\log{k}}$ for $k \ge 2$. Then let's find $\mathbb{E}|X| = \sum_{k \ge 2} \frac{k}{k^{2} \log{k}} = \infty$, so expectation doesn't converges.
Let's consider characteristic function of $X$ : $\phi(t) = \sum_{k\ge 2} \frac{e^{ikt}}{k^{2}\log{k}}$. So we have that our series is uniformly converges and function under sum sign is continuous , so we can find first derivate at zero. But when I differentiate it , I have that $\phi'(0)$ diverges. Where is my problem ? Does it have connection with complex part of function?
Where is the contradiction? If $\sum f_n(x)$ converges uniformly and each $f_n$ is differentiable it is not necessary that $\sum f'_n$ converges. Which theorem is being contradicted?