uniform convergence of Dirichlet eta function

Question

For any $x\in\mathbb{R}$, define a series $$\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{x}},$$ which is called the Dirichlet eta function from the complex analysis.

It converges pointwise on $(0,\infty)$ trivially.

I wonder about that what is the range for which converges uniformly?

By the little knowledge for complex analysis, it seems to be uniformly on any compact subset of $(0,\infty)$.

Is it possible to extend the range for uniform as $[\varepsilon,\infty)$? (where $\varepsilon>0$ is an arbitrary real number)

Give some advice. Thank you!

Answer 1

The series $\sum(-1)^{n}a_n$ converges if $a_n$ decreases to $0$. The proof of this theorem actually gives something stronger: if $f_n(x)$ decreases to $0$ uniformly for $x$ in some set $A$ then $\sum(-1)^{n}f_n(x)$ converges uniformly on $A$. From this it is obvious that the given series converge uniformly for $ x\in [\epsilon, \infty)$ [ since $\frac 1 {n^{x}} \leq \frac 1 {n^{\epsilon}}]$.

Answer 2

HINT:

Does $\frac1{n^x}$ converge uniformly to $0$ on $(0,1)$? What is the value of the limit $\lim_{n\to \infty}\frac{1}{n^{1/n}}$?