Evaluating $sum_{k=0}^{infty}frac{(-1)^k}{(k+1)^n}$


I got to evaluate this series, this is my approach
$$sum_{k=0}^{infty}frac{(-1)^k}{(k+1)^n}, text{ let } k+1=b, sum_{b=1}^{infty}frac{(-1)^{b-1}}{b^n}= -sum_{b=1}^{infty}frac{(-1)^{b}}{b^n} = -operatorname{Li}_n(-1)$$
But I’m not sure if this is how it’s done.
Any help is appreciated