Use the Ratio Test to determine whether each series converges absolutely or diverges. ∑n=2^∞ (3^

step 1 perform the limit ratio test and we need to compute the limit n goes to infinity of the am -dous

Use the Ratio Test to determine whether each series...

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Key Concepts

Absolute Convergence

Absolute convergence refers to a situation where the series formed by taking the absolute values of the terms of a given series also converges. If a series converges absolutely, then the original series converges regardless of the ordering of terms. This is a stronger form of convergence than conditional convergence and ensures that the series behaves well under rearrangements.

Ratio Test

The Ratio Test is a method used to determine the convergence or divergence of an infinite series. It involves computing the limit of the absolute value of the ratio of consecutive terms of the series. If this limit is less than one, the series converges absolutely; if it is greater than one, the series diverges; and if it equals one, the test is inconclusive. This test is especially useful when the series contains factorials, exponentials, or other rapidly growing functions.

Divergence

Divergence in the context of infinite series means that the series does not approach a finite limit as more terms are added. When applying convergence tests like the Ratio Test, if the limit of the ratio of successive terms is greater than one, it indicates that the terms of the series are not decreasing quickly enough, leading to divergence. Understanding divergence helps in classifying the behavior of series that do not settle towards any finite sum.

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