32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (√k / k − 1)²ᵏ

Verified step by step guidance

1. First, rewrite the general term of the series to understand its structure clearly. The term is given by \(a_k = \frac{\sqrt{k}}{k} - 1^{2k}\). Since the expression is ambiguous, clarify the term as \(a_k = \left( \frac{\sqrt{k}}{k} - 1 \right)^{2k}\) or \(a_k = \frac{\sqrt{k}}{k - 1^{2k}}\). Assuming the problem means \(a_k = \left( \frac{\sqrt{k}}{k} - 1 \right)^{2k}\), simplify the base inside the parentheses first.
2. Simplify the base inside the parentheses: \(\frac{\sqrt{k}}{k} = \frac{k^{1/2}}{k} = k^{-1/2}\). So the base becomes \(k^{-1/2} - 1\). Rewrite the term as \(a_k = \left( k^{-1/2} - 1 \right)^{2k}\).
3. Since the term is raised to the power \$2k$, consider using the Root Test or the Ratio Test to determine convergence. The Root Test is often simpler for terms raised to the power $k$. The Root Test involves computing \(\lim_{k \to \infty} \sqrt[2k]{|a_k|}\).
4. Calculate the \$2k$-th root of $|a_k|$: \(\sqrt[2k]{|a_k|} = |k^{-1/2} - 1|\). Then, find the limit as $k$ approaches infinity of this expression. This limit will help determine the behavior of the series.
5. Interpret the limit from the Root Test: if the limit is less than 1, the series converges absolutely; if it equals 1, the test is inconclusive; if greater than 1, the series diverges. Use this conclusion to classify the series as absolutely convergent, conditionally convergent, or divergent.

Key Concepts

Absolute and Conditional Convergence

A series converges absolutely if the series of absolute values converges. If the original series converges but not absolutely, it converges conditionally. Understanding these distinctions helps classify the behavior of infinite series.

Root Test

The Root Test evaluates the limit of the nth root of the absolute value of terms in a series. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive. It is especially useful for series with terms raised to the nth power.

Behavior of Terms Involving Powers and Radicals

Analyzing terms like (√k / k - 1) raised to the 2k power requires simplifying expressions involving radicals and powers. Understanding how these terms grow or shrink as k increases is crucial for applying convergence tests effectively.