Final answer:
A series converges if and only if it satisfies the Cauchy criterion, describing a sequence where terms get arbitrarily close to each other beyond a certain point.
Step-by-step explanation:
A series converges if and only if it satisfies the Cauchy criterion. This criterion essentially states that for the series to converge, the terms must get arbitrarily close to each other as the sequence progresses, in a manner where the difference between terms can be made as small as desired for all terms beyond a certain point. This concept is closely related to the notion of a Cauchy sequence, which is a sequence where the terms become arbitrarily close as their indices become sufficiently large. Another important criterion to note is the Divergence Test: if the limit of the terms of a series does not exist or does not equal zero, then the series must diverge.