9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
1 + (1 / 2)² + (1 / 3)³ + (1 / 4)⁴ + ⋯

Verified step by step guidance

1. Identify the general term of the series. The series is given as \$1 + \left(\frac{1}{2}\right)^2 + \left(\frac{1}{3}\right)^3 + \left(\frac{1}{4}\right)^4 + \cdots\(, so the \)n\(th term can be written as \)a_n = \left(\frac{1}{n}\right)^n$.
2. Recall the Root Test, which is often useful for series with terms raised to the power of \(n\). The Root Test states that for the series \(\sum a_n\), consider \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|}\). If \(L < 1\), the series converges absolutely; if \(L > 1\), it diverges; and if \(L = 1\), the test is inconclusive.
3. Apply the Root Test by computing \(\sqrt[n]{|a_n|} = \sqrt[n]{\left(\frac{1}{n}\right)^n} = \frac{1}{n}\).
4. Evaluate the limit \(L = \lim_{n \to \infty} \frac{1}{n}\). Since \(\frac{1}{n} \to 0\) as \(n \to \infty\), we have \(L = 0\).
5. Since \(L = 0 < 1\), by the Root Test, the series converges absolutely.

Key Concepts

Ratio Test

The Ratio Test determines the convergence of a series by examining the limit of the absolute value of the ratio of consecutive terms. 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.

Root Test

The Root Test analyzes convergence by taking the nth root of the absolute value of the nth term and finding its limit as n approaches infinity. 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.

Absolute Convergence

A series converges absolutely if the series of absolute values of its terms converges. Absolute convergence guarantees convergence of the original series and is a stronger condition than conditional convergence, simplifying the use of tests like the Ratio and Root Tests.