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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.7.27
Chapter 10, Problem 10.7.27

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 + ⋯

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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 \(a_n = \left(\frac{1}{n}\right)^n\).
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.
Apply the Root Test by computing \(\sqrt[n]{|a_n|} = \sqrt[n]{\left(\frac{1}{n}\right)^n} = \frac{1}{n}\).
Evaluate the limit \(L = \lim_{n \to \infty} \frac{1}{n}\). Since \(\frac{1}{n} \to 0\) as \(n \to \infty\), we have \(L = 0\).
Since \(L = 0 < 1\), by the Root Test, the series converges absolutely.

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Key Concepts

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

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.
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Ratio Test

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.
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Root Test

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.
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Choosing a Convergence Test
Related Practice
Textbook Question

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


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23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

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Textbook Question

55–70. More sequences

Find the limit of the following sequences or determine that the sequence diverges.


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Textbook Question

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.

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11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.


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Textbook Question

What comparison series would you use with the Comparison Test to determine whether ∑ (k = 1 to ∞) 2ᵏ / (3ᵏ + 1) converges?

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