Find the range of uniform convergence the Dirichlet series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^x}$ 
Find the range of uniform convergence of this Dirichlet series $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^x}.$$

I can apply the Weierstrass test to show the range is $1<x<\infty$. But I cannot show that at $x =1$ the series still converge uniformly. Can you tell me the formal proof of this?

The series converges uniformly for $x\ge a\gt0$ since The Alternating Series Test says
$$
\left|\sum_{k=m}^n\frac{(-1)^{k-1}}{k^x}\right|\le m^{-x}\le m^{-a}
$$
