Is there a way to meaningfully express $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}(x-1)^n}{n^k}$ for $k\neq 1$?

Question

For the following series: $$\sum_{n=1}^{\infty}\frac{\left(-1\right)^{\left(n+1\right)}\left(x-1\right)^{n}}{n^{k}}$$ other than $k=1$, is there any way to express this series as a function for other values of $k$?

Answer 1

Your series is related to the polylogarithm:

$$\sum_{n=1}^\infty \frac {(-1)^{n+1}(x-1)^n}{n^k}=-\sum_{n=1}^\infty\frac {(1-x)^n}{n^k}=-\operatorname{Li}_k(1-x)$$