11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ (k¹¹ + 2k⁵ + 1) / [4k(k¹⁰ + 1)]

Verified step by step guidance

1. Identify the general term of the series: \(a_k = ( -1 )^k \frac{k^{11} + 2k^5 + 1}{4k(k^{10} + 1)}\). Notice the factor \((-1)^k\) indicates this is an alternating series.
2. To apply the Alternating Series Test, focus on the absolute value of the terms without the alternating sign: \(b_k = \frac{k^{11} + 2k^5 + 1}{4k(k^{10} + 1)}\).
3. Simplify \(b_k\) by dividing numerator and denominator by the highest power of \(k\) in the denominator, which is \(k^{11}\), to analyze the limit as \(k \to \infty\).
4. Check the limit \(\lim_{k \to \infty} b_k\). If this limit is zero, proceed to the next step; otherwise, the series does not converge by the Alternating Series Test.
5. Verify if the sequence \(b_k\) is eventually decreasing for sufficiently large \(k\). If \(b_k\) is decreasing and the limit is zero, then by the Alternating Series Test, the series converges.

Key Concepts

Alternating Series Test

The Alternating Series Test determines the convergence of series whose terms alternate in sign. It requires that the absolute value of the terms decreases monotonically to zero. If these conditions hold, the series converges, even if it does not converge absolutely.

Behavior of the General Term

Analyzing the general term's behavior as k approaches infinity is crucial. Simplifying the term helps identify its limit and whether it approaches zero, a necessary condition for convergence of any infinite series.

Comparison and Simplification of Polynomial Expressions

Simplifying complex polynomial expressions in the numerator and denominator helps understand the dominant terms. This simplification aids in evaluating limits and determining the term's asymptotic behavior, which is essential for applying convergence tests.