Question

Find the limit of the sequence if it converges; otherwise indicate divergence.

7) an = ln(9n - 1) - ln(7n - 2)

Answer 1

The limit of the sequence an = ln(9n - 1) - ln(7n - 2) as n approaches infinity is the constant ln(9/7), indicating that the sequence converges to this value.

Step-by-step explanation:

The student is asking to find the limit of the sequence an = ln(9n - 1) - ln(7n - 2) as n approaches infinity. To find this limit, we can use the property that the logarithm of the quotient of two numbers is the difference between the logarithms of those two numbers. Therefore, we can rewrite the sequence as an = ln((9n - 1)/(7n - 2)).

As n approaches infinity, both (9n - 1) and (7n - 2) approach infinity as well, and the leading terms 9n and 7n will dominate the behavior of the sequence. Thus, the sequence behaves like ln(9n/7n) = ln(9/7). Since the logarithm of a constant is a constant, the limit of an as n approaches infinity is ln(9/7).

Thus, the sequence converges and the limit is ln(9/7).