syms rho L N_r
P_b = (1/2 * (1 - sqrt(rho/(2+rho)))^(2*N_r)) * symsum( nchoosek(2*N_r - 1 + L, L) * (1/2 * (1 - sqrt(rho/(2+rho)))^L), L, 1, 2*N_r - 1)
The result will still have a symsum() term when N_r is symbolic. If you subs() a particular numeric value in for N_r then it will be expanded.
There does not appear to be a nice compact formula involved; you just end up with a sum of terms.