The article is taking the first-order approximation of the doubling equation for discrete compounding interest, and as it turns out this value does not change n

The article is taking the first-order approximation of the doubling equation for discrete compounding interest, and as it turns out this value does not change no matter the compounding frequency. For different compounding frequencies, the validity of the approximation is governed by how close r/p is to 0. As p goes to infinity, the approximation becomes arbitrarily good, converging to the continuous case.