I am dealing with the convergence of the following series:
$ \sum_{n=1}^ \infty (x^n,(1-x)^n) $
I already showed that, though not sure, this series of functions converge pointwisely to the function $f(x)= ( \frac{1}{1-x} , \frac{1}{x} )$ on $(0,1)$, but diverges at $x=0,1$. I used the convergence (or divergence) of the real series $\sum_{n=1}^ \infty x^n $ and $\sum_{n=1}^ \infty (1-x)^n$. Is this correct?
More importantly, I want to argue that this convergence is uniform (or not) on certain subsets of $(0,1)$. I could not find a promising approach so far. I tried to measure the distance between the sequence of partial sums and the limit function using the supremum norm when arguing but maybe Weierstrass Test is more convenient?