Is the convergence rate of a series and its alternating counterpart same.

For example is the convergence rate of $\displaystyle \zeta(2)$ and $\displaystyle \eta(2)$ both equal.

What about the case $\displaystyle \zeta(1)$ and $\displaystyle \eta(1)$ where one is divergent and the latter is convergent to $\displaystyle \ln(2)$

Will the rates of convergence be equal to both $\displaystyle \zeta(s)$ and $\displaystyle \eta(s)$ for $\displaystyle s\geqslant 2$

Thanks