I was asked to check if the series of function $\sum_{n=0}^{\infty}(1-x^2)x^n$ uniformly convergences on $[0;1]$ or not.
This is my first try:
First, I find the partial sum: $S_n(x)=\sum_{k=0}^{n-1}(1-x^2)x^k=(1-x^2)\sum_{k=0}^{n-1}x^k=\begin{cases} 1 & \text{ if } x=0 \\ 0 & \text{ if } x=1 \\ (1-x^2)\frac{1-x^n}{1-x} & \text{ if } 0<x<1 \end{cases}=\begin{cases} 1 & \text{ if } x=0 \\ 0 & \text{ if } x=1 \\ (1+x)(1-x^n) & \text{ if } 0<x<1 \end{cases},$
we see that:
For $x=0$, $\forall n\in\mathbb{N}: S_n(0)=1$, so $\lim_{n\rightarrow\infty}S_n(0)=\lim_{n\rightarrow\infty}1=1 .$
For $x=1$, $\forall n\in\mathbb{N}:S_n(1)=0$, so $\lim_{n\rightarrow\infty}S_n(1)=\lim_{n\rightarrow\infty}0=0 .$
For $0<x<1, \forall n\in\mathbb{N}: S_n(x)=(1+x)(1-x^n)$, so $\lim_{n\rightarrow\infty}S_n(x)=\lim_{n\rightarrow\infty}(1+x)(1-x^n)=1+x.$
Thus, $S_n(x)$ converges pointwise to the function $S(x)=\begin{cases} 1 & \text{ if } x=0 \\ 0 & \text{ if } x=1\\ 1+x & \text{ if } 0<x<1 \end{cases} $ on $[0;1]$.
Now, we check if $S_n(x)$ uniformly converges to $S(x)$ on $[0;1]$. We have: $|S_n(x)-S(x)|=\begin{cases} |1-1| & \text{ if } x=0 \\ |0-0| & \text{ if } x=1\\ |(1+x)(1-x^n)-(1+x)| & \text{ if } 0<x<1 \end{cases} $ $=\begin{cases} 0 & \text{ if } x=0\\ 0 & \text{ if } x=1\\ (1+x)x^n & \text{ if } 0<x<1 \end{cases} $.
Thus, $\underset{x\in[0;1]}{sup}|S_n(x)-S(x)|=sup\left\{\underset{x\in(0;1)}{sup}[(1+x)x^n]; 0\right\}=sup\{2;0\}=2$. $\left(\underset{x\in(0;1)}{sup}(1+x)x^n=2 \: \text{since the function}\: g(x)=(1+x)x^n \: \text{is monotonically increasing on}\: (0;1),\:\text{so} (1+x)x^n<g(1)=2. \right)$
We see that $\underset{n\rightarrow\infty}{\lim}\underset{x\in[0;1]}{sup}|S_n(x)-S(x)|=\underset{n\rightarrow\infty}{\lim}2=2\neq 0,$ so $S_n(x)$ does not uniformly converge on $[0;1]$. And, so the given series of function $\sum_{n=0}^{\infty}(1-x^2)x^n$ does not uniformly convergences on $[0;1]$.
Is there any problems with my solution? Please, let me know where I should fix and modify!
It looks like I can't use the M-test in this case, so is there any other simpler ways to solve this problem! It's nice to see your recommendations! Thanks!