11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 2⁹k / kᵏ

Verified step by step guidance

1. First, write down the general term of the series: \(a_k = \frac{2^{9k}}{k^k}\).
2. To determine convergence, consider applying the Root Test, which is useful for series with terms raised to the power of \(k\). The Root Test uses the limit \(L = \lim_{k \to \infty} \sqrt[k]{|a_k|}\).
3. Calculate \(\sqrt[k]{|a_k|} = \sqrt[k]{\frac{2^{9k}}{k^k}} = \frac{2^9}{k}\), since \(\sqrt[k]{2^{9k}} = 2^9\) and \(\sqrt[k]{k^k} = k\).
4. Evaluate the limit \(L = \lim_{k \to \infty} \frac{2^9}{k}\). As \(k\) approaches infinity, \(\frac{2^9}{k}\) approaches 0.
5. Since \(L = 0 < 1\), by the Root Test, the series \(\sum_{k=1}^\infty \frac{2^{9k}}{k^k}\) converges absolutely.

Key Concepts

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining whether a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of the given series.

Root Test for Convergence

The Root Test involves taking the k-th root of the absolute value of the k-th term and examining its limit as k approaches infinity. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges. This test is particularly useful for series with terms raised to the k-th power.

Exponential and Factorial Growth Rates

Comparing growth rates of functions like exponentials and powers is crucial in convergence tests. In the series ∑ (2^(9k) / k^k), the denominator grows faster than any exponential due to k^k, which tends to infinity much faster, suggesting the terms approach zero rapidly, influencing convergence.