Final answer:
Both the subsequence of even terms and the subsequence of odd terms in a sequence converge to the same value s, so the entire sequence also converges to s. The correct answer is A. s.
Step-by-step explanation:
The question at hand is assessing understanding of sequences and their convergence properties in mathematics. When a sequence has a subsequence of even terms, denoted as (s2k), converging to a value s, and the subsequence of odd terms, (s2k+1), also converges to the same value s, we can infer that the entire sequence (sn) converges to that value as well. Since both the odd and even parts of the sequence are approaching s as their limit, the entire sequence must also approach s because every term in the sequence is getting arbitrarily close to s regardless of whether it is an odd or even term. Therefore, the full sequence (sn) converges to s.
The answer to the question, "If the subsequence of even terms (s2k) converges to s and the subsequence of odd terms (s2k+1) converges to s, then (sn) converges to __," is A. s.