Hello,
I'm new here and would like some help finding a function series qualifying the conditions in the title.
Thanks!
That's what he gave you. Let $\displaystyle g(x)=\sum_{n=1}^{\infty}\frac{(-1)^n}{n}$. This converges uniformly to $\displaystyle \ln\left(\frac{1}{2}\right)$ for a number of reasons...the easies to just state is Abel's Uniform Convergence Test -- from Wolfram MathWorld with $\displaystyle a_n=\frac{(-1)^n}{n},f_n(x)=1$.