function [tau] = Optimal_tau_func_pro3(k, gamma)
% This function calculates optimal tau by proposition 3.1
A = sum(gamma);
syms z
z_1 = solve( z*log(z)- z + 1 == A,z);
z = eval(z_1);
tau = zeros(1, k + 1);
tau(1) = (z - 1)/(A + z -1);
for i = 1 : k
    tau(i + 1) = gamma(i)/(A + z -1);
end