Does $\sum\limits_{k=1}^{n}\frac{(-1)^k}{k}(-1)^n$ converge?

The sequence $$u_n =\sum\limits_{k=1}^{n}\frac{(-1)^k}{k}$$

converges to $-\ln(2) < 0$. Therefore the sequence $$\sum\limits_{k=1}^{n}\frac{(-1)^k}{k}(-1)^n=(-1)^n u_n$$

does not converge.