Final answer:
The sequence converges if the terms approach a specific value as n approaches infinity. The fact given suggests that if the sequence converges to a limit L, then the adjacent terms in the sequence also approach L. To determine convergence, we analyze the limit of the sequence as n becomes very large.
Step-by-step explanation:
The question pertains to the convergence or divergence of sequences, a topic in mathematical analysis. By definition, a sequence {an} converges to a limit L if, as n approaches infinity, the terms of the sequence get arbitrarily close to L. The convergence of a sequence is dependent on the values of the terms as the index n becomes very large. A sequence diverges if it does not approach any particular value.
In practice, to determine if a given sequence converges, we often look at the limit of the sequence as n approaches infinity. If the limit exists and is finite, the sequence is said to converge; otherwise, it diverges. The provided fact that lim(n→[infinity]) an = lim(n→[infinity]) an+1 = lim(n→[infinity]) an−1 implies that if the sequence converges to a limit L, then this same limit L must also be shared by the adjacent members of the sequence (the next and the previous terms).