11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.


∑ (from k = 1 to ∞) (−k)³ / (3k³ + 2)

Verified step by step guidance

1. First, write down the general term of the series: \(a_k = \frac{(-k)^3}{3k^3 + 2}\).
2. Simplify the term \(a_k\): since \((-k)^3 = -k^3\), rewrite \(a_k\) as \(a_k = \frac{-k^3}{3k^3 + 2}\).
3. Analyze the behavior of \(a_k\) as \(k \to \infty\) by dividing numerator and denominator by \(k^3\): \(a_k = \frac{-1}{3 + \frac{2}{k^3}}\).
4. Determine the limit of \(a_k\) as \(k \to \infty\): \(\lim_{k \to \infty} a_k = \frac{-1}{3 + 0} = -\frac{1}{3}\).
5. Since the limit of \(a_k\) is not zero, conclude by the Test for Divergence (also called the nth-term test) that the series \(\sum_{k=1}^\infty a_k\) diverges.

Key Concepts

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates without settling.

Limit Comparison Test

The Limit Comparison Test compares a given series with a known benchmark series by examining the limit of their term ratios. If the limit is a positive finite number, both series either converge or diverge together, helping to determine the behavior of complex series.

Behavior of Polynomial Terms in Series

When series terms involve polynomials, the dominant powers in numerator and denominator dictate the term's behavior as k approaches infinity. Simplifying these terms helps identify if the series resembles a p-series or another known type, aiding in applying appropriate convergence tests.