Equal-rapidity chain of collinear Lorentz boosts ($v_N=c\tanh (N\mathrm{artanh}(u/c))$)

If each successive inertial frame moves at the same signed velocity $u$, with $|u|<c$, relative to the preceding frame along a common axis, the rapidity adds. The final velocity relative to the first frame is therefore the displayed expression. When $u=c/N$ and $N\ge 2$, the limit is $c\tanh 1$, since $N\mathrm{artanh}(1/N)\to 1$.