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