Uniform convergence of $\sum (-1)^nf_n(x)$ on $[0,1]$ where $f_n(x)=x^n(1-x)$.

Let $f_n(x) = x^n(1-x)$ and $\sum (-1)^nf_n(x)$.

I showed that this series point wise.

Case1) $x=1$ $$ f_n(1)=0 \, , \sum{(-1)^nf_n(1)} = 0 $$ Case2) $x\neq 1$ $$|f_n(x)| \leq |x|^n$$ since $$\sum |x|^n$$ converges, By abosultely convergese test $\sum (-1)f_n(x)$ converges.

I want to show that this series uniformly converges on $[0,1]$.

Solution 1:

Hint. One may recall that $$ \sum_{n=0}^Nx^n=\frac{1-x^{N+1}}{1-x}, \quad x\neq1, $$ giving $$ \sum_{n=0}^N(-1)^nx^n=\frac{1+(-1)^Nx^{N+1}}{1+x}, \quad 0\leq x\leq1, $$ then

$$ \left|\sum_{n=0}^N(-1)^nx^n-\frac{1}{1+x}\right|\leq\frac{|x|^{N+1}}{1+x}\leq |x|^N, \quad 0\leq x<1 $$

may help to conclude.