I seek advice on whether a certain very long function can be made significantly more compact and therefore easier to compute.
I am interested in calculating the value of a function which resembles both a continued fraction and an iterated function, where the basic unit of the iteration is the right hand side of the formula for relativistic addition of two (collinear) velocities:
(1) W_1 = (u + v) / (1+uv) .
u and v are the summed velocities, given as decimal fractions of the velocity of light C. Ie: u,v range from 0.0 to 1.0 . Their resultant velocity is W_n.
The essential idea is to divide C into n <equal> velocities (v) and then add them sequentially. My procedure is to take the first sum, W, and then add to it a third identical velocity v, which gives a second resultant velocity
(2) W_2 = (W_1 + v) / (1 + vW_1) .
This is an iteration of the basic form in (1).
Similarly, a fourth identical velocity v is added to W_2, giving
(3) W_3 = (W_2 + v) / (1+ vW_2) ,
and so on and on...
This is to be repeated n times, where n becomes very large and v commensurately smaller. I wish to know the values of this function as n becomes extremely large. I especially want to know what the function converges to as n ---> infinity and C/n ---> zero velocity.
A partial glimpse of this procedure as a single large relation resembles an unusual variety of continued fraction. I omit rendering it in ASC II form here since it rapidly becomes very confusing and opaque.