- The series $\sum_{n= 0}^{\infty}x^n\ln(x)$ converges pointwise on $(0,1]$ to $f(x)=\begin{cases}\frac{\ln x}{1-x} & \text{if}\ x\in (0,1)\\ 0 & \text{if}\ x=1 \end{cases}$.
For any fixed positive integer $N$, since the function $S_N(x)=\sum_{n= 1}^Nx^n\ln(x)$ is continuous on $(0,1]$ but $f$ is not continuous as $\lim\limits_{x\to 1}f(x)=-1\neq 0=f(1)$, we do not have uniform convergence.
- I cannot decide whether $\sum_{k= 1}^{\infty}x^n(\ln(x))^2$ converges uniformly on $(0,1]$ to $g(x)=\begin{cases}\frac{(\ln x)^2}{1-x} & \text{if}\ x\in (0,1)\\ 0 & \text{if}\ x=1 \end{cases}$ or not. The above argument does not work since $g$ is continuous at $x=1$. Finding $\sup\limits_{x\in (0,1]}|S_N(x)-g(x)|$ is also difficult.
Did I do 1. correctly? Any hints for 2.?
Hints for 2. i) What is the maximum of $x^n(\ln x)^2$ on $(0,1]?$ ii) Weierstrass M-test.