Given is the function $f_k(x)=\tan^k(x)$ for $x\in\mathbb{R}$ except for $x=\frac{\pi}{2}+j\pi\ (j\in\mathbb{Z})$, where $f_k(x)=0$. I want to know in which points $x\in\mathbb{R}$ the series $\sum_{k\geqslant 0}f_k$ converges pointwise and what the limit function $g$ is. Also, I want to know on which intervals $I\subset\mathbb{R}$ the series converges uniformly to $g$.
For $x=\frac{\pi}{2}+j\pi$ it is easy to see that $g=0$, so the sequence $(F_n)=\sum_{k=0}^n f_k$ of partial sums converges for this $x$. So this is also true for the series. Know, when $-\pi/4\leqslant x\leqslant \pi/4$, I know that $-1\leqslant \tan^k(x)\leqslant 1$. But then the limit $g=\lim_{n\rightarrow\infty}F_n(x)=\lim_{n\rightarrow\infty}\sum_{k=0}^\infty f_k\leqslant1+1+\dots$, but this diverges. What am I doing wrong?
Generally, $\sum_{k\geq 0}q^k$ converges iff $|q|\leq 1$, and in this case the limit is $\frac{1}{1-q}$. Use this with $q=\tan x$ to determine the domain of convergence of your series.
Remark: $\tan(\frac{\pi}{2}+j\pi)$ is not defined (or one can say it is $\infty$).