Using the Root Test
In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.
∑(from n=1 to ∞) [4ⁿ / (3n)ⁿ]

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Identify the general term of the series: \(a_n = \frac{4^n}{(3n)^n}\).

Recall the Root Test formula: compute \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|}\).

Apply the nth root to the term: \(\sqrt[n]{|a_n|} = \sqrt[n]{\frac{4^n}{(3n)^n}} = \frac{\sqrt[n]{4^n}}{\sqrt[n]{(3n)^n}}\).

Simplify the nth roots: \(\frac{4}{3n}\).

Evaluate the limit as \(n\) approaches infinity: \(L = \lim_{n \to \infty} \frac{4}{3n}\). Use this value to determine convergence: if \(L < 1\), the series converges absolutely; if \(L > 1\), it diverges; if \(L = 1\), the test is inconclusive.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Root Test for Series Convergence

The Root Test determines the convergence of an infinite series by examining the nth root of the absolute value of its terms. Specifically, if the limit of the nth root of |a_n| as n approaches infinity is less than 1, the series converges absolutely; if greater than 1, it diverges; and if equal to 1, the test is inconclusive.

Absolute Convergence

A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence guarantees convergence regardless of the sign of terms, making it a stronger form of convergence and simplifying the analysis of series with alternating or complex terms.

Evaluating Limits Involving nth Roots

To apply the Root Test, one must compute the limit of the nth root of the series terms. This often involves simplifying expressions with powers and roots, such as rewriting terms like (4^n / (3n)^n)^(1/n) into a more manageable form to find the limit as n approaches infinity.

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