In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 1 to ∞) [3ⁿ / n³]

Verified step by step guidance

1. Identify the given series: \( \sum_{n=1}^{\infty} \frac{3^n}{n^3} \). This is an infinite series with terms \( a_n = \frac{3^n}{n^3} \).
2. Recognize that the numerator \(3^n\) is an exponential function and the denominator \(n^3\) is a polynomial function. Generally, exponential growth dominates polynomial growth as \(n\) becomes large.
3. Consider applying the Root Test or Ratio Test, which are effective for series involving exponential terms. For the Ratio Test, compute \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
4. Calculate \( \frac{a_{n+1}}{a_n} = \frac{3^{n+1} / (n+1)^3}{3^n / n^3} = 3 \cdot \frac{n^3}{(n+1)^3} \). Then find the limit \( L = \lim_{n \to \infty} 3 \cdot \frac{n^3}{(n+1)^3} \).
5. Evaluate the limit \( L \). If \( L > 1 \), the series diverges; if \( L < 1 \), the series converges; if \( L = 1 \), the test is inconclusive. Use this conclusion to determine the behavior of the series.

Key Concepts

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining convergence means checking if the sum approaches a finite limit as the number of terms grows. If the series converges, its partial sums get closer to a specific value; if it diverges, the sums grow without bound or oscillate.

Ratio Test

The Ratio Test is used to determine the convergence of series with positive terms, especially those involving exponentials or factorials. It examines the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges.

Comparison of Growth Rates (Exponential vs Polynomial)

In series terms, exponential functions grow faster than any polynomial function as n approaches infinity. When the numerator grows exponentially and the denominator polynomially, the terms often do not approach zero, leading to divergence. Recognizing this helps in quickly assessing series behavior.