Victor Protsak formula root of n-th degree simple polynomial

(*start*)
Clear[x, k];
n = 6;
a = 1/3;
N[Sum[Binomial[n*k, k]*a^((n - 1)*k + 1)/((n - 1)*k + 1), {k, 0,
   2000}], 20]
x = N[Sum[
   Binomial[n*k, k]*a^((n - 1)*k + 1)/((n - 1)*k + 1), {k, 0, 3000}],
  20]
Print["If this is zero then x is solution to the equation:"]
x^n - x + a
(*end*)