Cauchy convergence test

The Cauchy convergence test checks if a sequence is approaching a specific value, called the limit, without needing to know that limit beforehand. It says that if, for any small distance you choose, the terms of the sequence eventually become closer to each other than that distance, then the sequence is converging. Essentially, as you go further out in the sequence, the numbers get increasingly close to each other, which strongly indicates they’re approaching a single point. This method is useful because it allows us to confirm convergence based only on the behavior of the sequence itself, without knowing the limit ahead of time.