$\sum_{n=1}^\infty x(1-x)^{n-1}$ Does this sum converge uniformly?

Question

$$\sum_{n=1}^\infty x(1-x)^{n-1}$$

I know that this sum converge $\iff$ $0\le x \le 1$, i wanted to use the Weierstrass but could not suceed, so i think this sum might not converge uniformly,but i'm having problem showing it.

Answer 1

1. The series $\sum_{n=1}^\infty x(1-x)^{n-1}$ converges for $0 \le x <2$ !!

2. Show that $\sum_{n=1}^\infty x(1-x)^{n-1}=1$ for all $x \in (0,2)$.

3. Let $s_N(x):=\sum_{n=1}^N x(1-x)^{n-1}$ and show that $|s_N(x)-1|=|1-x|^N$.

4. Let $x_N:=1-\frac{1}{2^{1/N}}$ and show that $|s_N(x_N)-1| =1/2$ for all $N$.

5. Conclude from 4. that the series does not converge uniformly.

Answer 2

Let $f(x)=\sum_{n=1}^\infty x(1-x)^{n-1}$ for $x \in [0,2)$. Then

$f(x)=1$ if $x \in (0,2)$ and $f(0)=0$. Hence $f$ is not continuous on $[0,2)$. Therefore the series does not converge uniformly.