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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.1.55
Chapter 10, Problem 10.1.55

Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = 8^(1/n)

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Identify the given sequence: \(a_n = 8^{\frac{1}{n}}\).
Recall that as \(n\) approaches infinity, the term \(\frac{1}{n}\) approaches 0.
Rewrite the sequence in terms of an exponential function: \(a_n = e^{\ln(8^{\frac{1}{n}})} = e^{\frac{1}{n} \ln(8)}\).
Analyze the limit of the exponent as \(n \to \infty\): \(\lim_{n \to \infty} \frac{1}{n} \ln(8) = 0\) because \(\ln(8)\) is a constant and \(\frac{1}{n} \to 0\).
Use the continuity of the exponential function to find the limit of the sequence: \(\lim_{n \to \infty} a_n = e^0 = 1\). Therefore, the sequence converges to 1.

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Key Concepts

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

Sequence and Limit

A sequence is an ordered list of numbers defined by a formula for its nth term. The limit of a sequence is the value that the terms approach as n becomes very large. Understanding how to find and interpret limits is essential to determine if a sequence converges or diverges.
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Convergence and Divergence of Sequences

A sequence converges if its terms approach a finite number as n approaches infinity; otherwise, it diverges. Recognizing whether a sequence converges or diverges helps in analyzing its long-term behavior and is fundamental in calculus.
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Properties of Exponential and Root Functions

Expressions like 8^(1/n) involve roots and exponents. As n increases, the nth root of a positive number approaches 1. Understanding these properties allows you to evaluate the limit of sequences involving roots and powers effectively.
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Related Practice
Textbook Question

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) (tanh n) / n²

Textbook Question

In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.

∑ (from n = 0 to ∞) [(x + 1)²ⁿ] / 9ⁿ

Textbook Question

Direct Comparison Test

In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.

∑ (from n=1 to ∞) cos²n / n^(3/2)

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Textbook Question

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = (2n + 2)! / (2n − 1)!

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Textbook Question

Absolute and Conditional Convergence

Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ⁻¹ / (n² + 2n + 1)]

Textbook Question

Suppose that aₙ > 0 and limₙ→∞ n²aₙ = 0. Prove that ∑aₙ converges.

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