Perhaps it would be good to get back to the original problem which was the suspicion that a "sum of sinusoids" and a "square-wave" are fundamentally different. Just because the shapes are not similar does not allow you to conclude that they are fundamentally different. It is of course correct to say what level of precision you require for equality. Transcendental functions by their very nature are imprecise having both irrational arguments and irrational values. In a sense nothing involving transcendental functions can ever be precise in the sense that a rational result can be.
If there is some difference between the two things would the OP please enlighten us with a convincing argument to support that conjecture.
If there is some difference between the two things would the OP please enlighten us with a convincing argument to support that conjecture.