series: $S(x)=\sum_{n=0}^\infty x^n(1-x)^2$ is NOT uniformly convergent in $[0,1]$

Question

Question asks :

To show that the series: $S(x)=\sum_{n=0}^\infty x^n(1-x)^2$ is NOT uniformly convergent in $[0,1]$

My work :

sum function $S(x) = 1-x \ ,$ $\forall$ $x$ $\in$ $[0,1]$

By Dini's Theorem which reads as follows as per my book "If the sum function of a series $\sum f_{n}$ with non-negative continuous terms defined on an interval $[a,b]$ is continuous on $[a,b]$, then the series is uniformly convergent on the interval"

I find this series to be uniformly convergent. Kindly help what am i missing

Answer 1

The series is uniformly convergent on $[0,1]$.

The maximum of $f_n(x) = (1-x)^2x^n$ can be found by setting the derivative to zero and is attained at $x = n/(2+n)$. Thus, for $x \in [0,1]$

$$0 \leqslant f_n(x) \leqslant \left(\frac{2}{2+n}\right)^2 \left(\frac{n}{2+n}\right)^n< \frac{4}{(2+n)^2} < \frac{4}{n^2}$$

Uniform convergence follows from the Weierstrass test.