I'm having trouble proving that the function below is continuous on [1,e]
$$ \sum_{n=1}^\infty \frac{1}{n3^n} ln(x) $$
Many answers I've seen already involve derivatives (which I cannot use) or just appear to have incomplete or incorrect answers.
Any help outlining a proof? I'm allowed to use the fact that the natural logarithm is continuous and increasing without proof.