Consider $f_n(x) = \sum_{k=0}^{n} {x^k}$. Does $f_n$ converge pointwise on $[0,1]$? Does it converge uniformly on $[0,1]$?
Well, my approach would be, first of all to notice that if $x \neq 1$, then by simple induction we get $f_n(x) = \frac{1 - x^{n+1}}{1-x}$
So, $(f_n) \rightarrow \frac{1}{1-x}$. But at $x = 1$, the $f_n \rightarrow \infty$, hence $(f_n)$ does not converge pointwise. Therefore, it does not converge uniformly. Is this correct? Hope to get feedback
thanks,