Iterated collinear boosts with equal rapidity ($u_n=c\tanh (n\alpha )$)

The relativistic velocity-addition formula becomes addition of rapidities for collinear Lorentz boosts. Starting from rest, applying the same positive rest-frame velocity increment $c\tanh \alpha$ repeatedly gives $u_n=c\tanh (n\alpha )$. For increment $c/2$, $\alpha =\frac{1}{2}\log 3$ and $u_n=c(3^n-1)/(3^n+1)$. Every finite sequence remains below the speed of light.