The function $ \frac{-\gamma-ln(1-x)}{1-x} $ has series expansion: $$ \psi(1)+\psi(2)x+\psi(3)x^2+... $$ where $\psi(x)$ is the digamma function and $\gamma$ is the Euler-Mascheroni constant. Note that $\psi(x) \approx ln(x)$. This raises the question of what function the series: $$ ln(1)+ln(2)x+ln(3)x^2+... $$ converges to.
Another interesting question this raises is whether there exists a function $f(x)$ such that $ \frac{-\gamma-f(1-x)}{1-x} $ has series expansion: $$ f(1)+f(2)x+f(3)x^2+... $$ Does there exist such a function? What other properties does this function have? Are there other functions that follow a similar rule?
By Frullani's theorem $$ \log(n) = \int_{0}^{+\infty}\frac{e^{-t}-e^{-nt}}{t}\,dt $$ hence for any $x$ such that $|x|<1$ we have $$ \sum_{n\geq 1}\log(n) x^n = \int_{0}^{+\infty}\sum_{n\geq 1}\frac{e^{-t}-e^{-nt}}{t}x^n\,dt = \frac{x^2}{1-x}\int_{0}^{+\infty}\frac{1-e^{-t}}{t}\cdot\frac{dt}{e^t-x} $$ which is not an elementary function, but it has a pretty compact integral representation, allowing accurate numerical approximations. The same trick can be performed for $\sum_{n\geq 1}f(n)\,x^n$ if $f$ has a manageable inverse Laplace transform.