Final answer:
Applying the Ratio Test to the series ∑ₙ₌₁^(∞) (1.01)^ₙ / ₙ^6, we examine the limit of the ratio of consecutive terms as n approaches infinity. Upon evaluation, the limit of the absolute value of the ratio is less than 1. Therefore, by the Ratio Test, the series converges.
Step-by-step explanation:
The Ratio Test is a tool used to determine the convergence or divergence of an infinite series involving the ratio of consecutive terms. For the series ∑ₙ₌₁^(∞) (1.01)^ₙ / ₙ^6, the Ratio Test involves finding the limit as n approaches infinity of the absolute value of the ratio of successive terms: limₙ→∞ |aₙ₊₁ / aₙ|. Here, aₙ represents the nth term of the series.
In this case, for the given series ∑ₙ₌₁^(∞) (1.01)^ₙ / ₙ^6, the ratio of consecutive terms is (1.01)^(n₊₁) / (n₊₁)^6 divided by (1.01)^ₙ / ₙ^6. When simplifying this expression, it leads to (1.01) * (n^6) / ((n₊₁)^6). As n approaches infinity, this ratio tends to (1.01) * 1, which is 1.01. Since the limit of the ratio is less than 1, the Ratio Test concludes that the series converges.
Therefore, by the Ratio Test, which ensures convergence when the limit of the absolute value of the ratio of consecutive terms is less than 1, the series ∑ₙ₌₁^(∞) (1.01)^ₙ / ₙ^6 converges. This implies that as more terms are added to the series, the sum approaches a finite value rather than diverging towards infinity.