Final answer:
The given sequence (2 · 4 · 6 · · · · ·(2n) · 1) / (1 · 3 · 5 · · · · · (2n − 1) · n^2) converges to 1.
Step-by-step explanation:
The given sequence is (2 · 4 · 6 · · · · ·(2n) · 1) / (1 · 3 · 5 · · · · · (2n − 1) · n^2).
To determine if the sequence converges, we can simplify the expression. By factoring out a 2 from the numerator and a n from the denominator, the expression becomes:
(2/n) · (1 · 2 · 3 · · · · ·(2n-1) · n) / (1 · 3 · 5 · · · · · (2n − 1) · n).
Notice that the terms in the numerator and denominator are the same, except for the first term (2/n). As n approaches infinity, the first term becomes closer to 0. Therefore, the entire expression approaches 1 as n increases. So, yes, the sequence converges to 1.