11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (2k⁴ + k) / (4k⁴ − 8k)

Verified step by step guidance

1. First, write down the general term of the series: \(a_k = \frac{2k^4 + k}{4k^4 - 8k}\).
2. To analyze convergence, consider the behavior of \(a_k\) as \(k\) approaches infinity. Simplify the expression by dividing numerator and denominator by the highest power of \(k\) present in the denominator, which is \(k^4\):
3. \[a_k = \frac{2k^4 + k}{4k^4 - 8k} = \frac{2 + \frac{1}{k^3}}{4 - \frac{8}{k^3}}.\]
4. Evaluate the limit of \(a_k\) as \(k \to \infty\): \(\lim_{k \to \infty} a_k = \frac{2 + 0}{4 - 0} = \frac{2}{4} = \frac{1}{2}\). Since this limit is not zero, the terms do not approach zero.
5. Recall the necessary condition for series convergence: if \(\lim_{k \to \infty} a_k \neq 0\), then the series \(\sum a_k\) diverges. Therefore, conclude that the series diverges.

Key Concepts

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Determining convergence involves analyzing the behavior of the terms as the index grows large, ensuring the sum does not diverge to infinity or oscillate indefinitely.

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 this limit is a positive finite number, both series either converge or diverge together, making it useful for series with rational expressions.

Behavior of Rational Functions for Large k

For large values of k, the dominant terms in the numerator and denominator of a rational function determine its behavior. Simplifying by focusing on highest-degree terms helps approximate the general term, which is essential for applying convergence tests effectively.