sum polylogarith convergence

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I am working in the following series i get a result could you help to see the convergence any ideas $$\sum _{k=1}^{\infty } \frac{(-1)^{k+1} \text{Li}_k(-a)}{a}=\frac{a-\log (a+1)}{a^2}$$ where Lk is the polylogaritm.