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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 10.1.31
Chapter 10, Problem 10.1.31

Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = 2 + (0.1)ⁿ

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1
Identify the general term of the sequence: \(a_n = 2 + (0.1)^n\).
Recall that a sequence converges if its terms approach a finite limit as \(n\) approaches infinity.
Analyze the behavior of the term \((0.1)^n\) as \(n \to \infty\). Since \(0.1\) is a number between 0 and 1, \((0.1)^n\) approaches 0 as \(n\) becomes very large.
Use the limit laws to find the limit of the sequence: \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left( 2 + (0.1)^n \right) = 2 + \lim_{n \to \infty} (0.1)^n\).
Since \(\lim_{n \to \infty} (0.1)^n = 0\), conclude that \(\lim_{n \to \infty} a_n = 2 + 0 = 2\), so the sequence converges to 2.

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Key Concepts

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

Sequence Convergence

A sequence converges if its terms approach a specific finite value as n approaches infinity. This limit is the value the sequence gets arbitrarily close to, indicating stability in the sequence's behavior over time.
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Limit of a Sequence

The limit of a sequence {aₙ} is the value L that the terms aₙ approach as n becomes very large. If such a limit exists and is finite, the sequence converges to L; otherwise, it diverges.
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Behavior of Exponential Terms with Base Between 0 and 1

For a term like (0.1)ⁿ, where the base is between 0 and 1, the value approaches zero as n increases. This property helps determine the limit of sequences involving such terms.
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Related Practice
Textbook Question

Determining Convergence or Divergence

In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑(from n=1 to ∞) [(-1)ⁿ n² e⁻ⁿ]

Textbook Question

Finding Taylor Polynomials

In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.

f(x) = √(1 − x),a = 0

Textbook Question

Determining Convergence or Divergence

In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]

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Textbook Question

Convergence or Divergence

Which of the series in Exercises 57–64 converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ (n!)ⁿ] / [n^(n²)]

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Textbook Question

In Exercises 53–56, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.

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

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 ∞) 1 / (2√n + ³√n)

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