= Iterated collinear boosts with equal rapidity
{title2=$u_n=c\tanh(n\alpha)$}
Starting from rest, compose $n$ positive collinear <Lorentz boosts>, each with the same finite <rapidity> $\alpha>0$. Each step has relative <velocity> $w=c\tanh\alpha$, measured in the preceding rest <inertial frame>. The <relativistic velocity-addition formula> gives
$$
u_0=0,\qquad u_{n+1}=\frac{u_n+w}{1+u_nw/c^2},\qquad \theta_n=n\alpha,\qquad u_n=c\tanh(n\alpha).
$$
These are equal increments of <rapidity>, not equal increments of <velocity> measured in the original <inertial frame>. They can describe a chain of relatively moving <inertial frames> or successive idealized changes of a massive particle's motion; no particular acceleration schedule is assumed. https://davidtong.org/teaching/general-relativity/grhtml/S1[David Tong, General Relativity, §1.2.4: collinear boost composition and rapidity]{ref}.
For a relative <velocity> $w=c/2$ at each step,
$$
\alpha=\operatorname{artanh}(1/2)=\frac12\ln3,\qquad \frac{u_n}{c}=\frac{3^n-1}{3^n+1}.
$$
The first four <velocities> are $u_0=0$, $u_1=c/2$, $u_2=4c/5$, and $u_3=13c/14$. At step eight, $u_8/c=3280/3281\approx0.99969521$. The <rapidities> grow by the same amount at each step, while the successive <velocity> increments become smaller.
\Image[/rapidity-collinear-boosts.png]
{title=Equal collinear boosts add rapidities linearly while their composed velocities approach the speed of light}
{id=image-rapidity-collinear-boosts}
{description=The illustrative sequence uses relative <velocity> $c/2$ per step and $n=0,\ldots,8$. The horizontal coordinate counts boosts, rather than elapsed time. The line at $u_n/c=1$ marks the <speed of light>; every plotted <velocity> is strictly below it.}
A short derivation applies the <hyperbolic-function identity> for $\tanh(a+b)$ to the recurrence, giving $\theta_{n+1}=\theta_n+\alpha$. Substituting $\tanh x=(e^{2x}-1)/(e^{2x}+1)$ and $e^{2\alpha}=3$ gives the displayed closed expression. In particular,
$$
1-\frac{u_n}{c}=\frac{2}{3^n+1}>0
$$
for every finite nonnegative integer $n$. The <limit> is $u_n\to c$ as $n\to\infty$, but no finite sequence reaches the <speed of light>. This illustrates the beauty of <rapidity>: unbounded additive parameters coexist with bounded physical <velocities>. Replacing every step by a negative <rapidity> gives the corresponding approach to $-c$.
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