## Convergence rate of series

Is the convergence rate of a series and its alternating counterpart same.
For example is the convergence rate of $\zeta(2)$ and $\eta(2)$ both equal.

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

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

Thanks