Consider the sequence of functions $f_n:[0,1]$,$f_n:=(-1)^n(1-x)x^n, n=0,1,2,....,0<=x<=1.$
a. Compute the sum of the series $f(x):=\sum_{n=0}^{\infty}f_n(x),$ x belongs to [0,1].
b. Show that the series $f(x):=\sum_{n=0}^{\infty}f_n(x)$ converges to $f$ uniformly on [0,1].
I've been having difficulty solving this problem and would appreciate any help!
Part a) involves evaluation of geometric series:
$$\sum_{n=0}^{\infty} f_n(x) = (1-x) \sum_{n=0}^{\infty} (-1)^n x^n = \frac{1-x}{1+x}$$