Solution
= Solution
Take $v>0$ for recession. By the definition of <kinematic redshift>, $1+z_{\rm kin}=\nu_e/\nu_0$. Inverting the supplied <relativistic Doppler effect> and expanding with $\beta=v/c$ gives
$$
1+z_{\rm kin}=\sqrt{\frac{1+\beta}{1-\beta}}
=(1+\tfrac12\beta+O(\beta^2))(1+\tfrac12\beta+O(\beta^2))
=1+\beta+O(\beta^2).
$$
Thus
$$
\boxed{z_{\rm kin}\simeq v/c\quad\text{for }|v|\ll c.}
$$
A negative radial <velocity> gives a blueshift at this order.