Relationship between partial sum of Riemann zeta functions over even integers and the harmonic series

Question

How do you prove that ${\sum_{n=1}^{k}(\zeta(2*n)/n)-H_k(1)}$ tends to $\ln(2)$ as integer $k$ tends to infinity where $H_k(1) = \sum_{n=1}^{k}{1\over n}$? Is this result well known? Please give a reference if it is. $\zeta(x)$ is the Riemann zeta function evaluated at $x$. I have a proof.