Low bound of Dirichlet eta function

Question

every one. Suppose that $\eta(s)$ is Dirichlet eta function, I may find a low bound of this function, namely

$\eta(2n)>\frac{2^{2n-1}-2}{2^{2n-1}-1}$

with $n>1$ and $n$ is a integer. But is this true? can somebody prove or disprove it?

Answer 1

Well, in fact, one can use the fact that

$1+\frac{1}{2^x}+\int_2^{\infty}\frac{1}{n^x}dx>\sum_{n=1}^{\infty}\frac{1}{n^x}>1+\frac{1}{2^x}+\int_3^{\infty}\frac{1}{n^x}dx$.

The inequality chain gives a stronger form.