Final answer:
The divergence test involves finding the limit of the general term of the series as n approaches infinity. For the series ∑ln(8n)/9^n, this limit is zero, which makes the divergence test inconclusive. Additional tests are required to determine convergence or divergence.
Step-by-step explanation:
The purpose of the divergence test is to determine whether a given series converges or diverges. In this case, we're looking at the series ∑n=2∞ln(8n) / 9n. To apply the divergence test, you calculate the limit as n approaches infinity of the general term of the series. If this limit is not zero, then the series diverges.
To find this limit, consider the term ln(8n) / 9n. As n approaches infinity, the numerator, ln(8n), grows without bound but at a much slower rate than the denominator, 9n, which grows exponentially. Applying L'Hôpital's Rule repeatedly would show that the numerator's growth rate is not sufficient to outpace the exponential growth of the denominator, and thus the limit is zero:
∑n→∞lim (ln(8n)/9n) = 0
However, this does not mean the series converges. The divergence test is inconclusive in this case because it only indicates that the series might converge; it does not provide a definitive answer. To fully determine convergence or divergence, further tests like the Ratio Test, Root Test, or comparison with a known convergent series should be applied.