Final answer:
To demonstrate that the sequence (3n - 1)/(2n + 1) is convergent, we consider the limit as n approaches infinity and note that the ratio of the leading coefficients (3/2) shows the sequence converges to 3/2.
Step-by-step explanation:
To show that the sequence (3n - 1)/(2n + 1) is convergent, we use the fact that as n approaches infinity, the highest degree terms in the numerator and the denominator dominate the behavior of the sequence. In this case, both the numerator and the denominator have the same degree terms (n). Therefore, the limit can be found by dividing the coefficients of the highest degree terms.
The limit as n approaches infinity of (3n - 1)/(2n + 1) equals the limit as n approaches infinity of 3n/2n, because the -1 in the numerator and the +1 in the denominator become insignificant as n grows without bounds. Thus, this sequence has a limit of 3/2.
This proves that the sequence is convergent, as it approaches a specific value, which in this case is 3/2.