Uniform convergence and sequence of functions I was asked to prove that $f_n(x) = (1-x)^{\alpha}x^n$ is uniformly convergent on $[0,1]$ where $\alpha&gt;0$ I did this part then I was asked if wether the corresponding series of functions converges uniformly on the given interval ? And if it does for what alpha it should be ?
   Hints are really appreciated ? Thanks
Let $S_n(x)=\sum _{k=0}^n(1-x)^\alpha x^k=(1-x)^\alpha \sum_{k=0}^nx^k=(1-x)^\alpha\dfrac{1-x^n}{1-x}=(1-x)^{\alpha-1}(1-x^n)$

$S(x)=\lim_{n\to\infty}S_n(x)=(1-x)^{\alpha-1}(1-x^n)$ which can be checked to be discontinuous irrespective of $\alpha$
Let $S_n(x)=\sum _{k=0}^n(1-x)^\alpha x^k=(1-x)^\alpha \sum_{k=0}^nx^k=(1-x)^\alpha\dfrac{1-x^n}{1-x}=(1-x)^{\alpha-1}(1-x^n)$

$S(x)=\lim_{n\to\infty}S_n(x)=(1-x)^{\alpha-1}(1-x^n)$ which can be checked to be discontinuous irrespective of $\alpha$