Final answer:
To determine whether the sequence an = 3n² + 9n + 1 is convergent, divide every term by n² and calculate the limit as n approaches infinity. If the limit exists and is finite, the sequence is convergent. In this case, the sequence is convergent with a limit of 12.
Step-by-step explanation:
To determine whether the sequence an = 3n² + 9n + 1 is convergent, we need to check if the terms of the sequence approach a specific value as n approaches infinity. One way to do this is to find the limit of the sequence as n goes to infinity. If the limit exists and is finite, then the sequence is convergent.
To find the limit of the sequence an = 3n² + 9n + 1 as n approaches infinity, we can divide every term by n². This gives us:
lim(an/n²) = lim(3n²/n²) + lim(9n/n²) + lim(1/n²)
As n goes to infinity, the first and second terms simplify to 3 and 9, respectively. The third term becomes 0. Therefore, the limit of the sequence is 3 + 9 + 0 = 12.
Since the limit exists and is finite (12), we can conclude that the sequence an = 3n² + 9n + 1 is convergent.