Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.2.67

Chapter 10, Problem 10.2.67

55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.

{sinn / 2ⁿ}

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Identify the given sequence as \( a_n = \frac{\sin n}{2^n} \), where \( n \) is a positive integer.

Recall that \( \sin n \) oscillates between -1 and 1 for all integer values of \( n \), so \( \sin n \) is bounded.

Note that the denominator \( 2^n \) is an exponential function that grows without bound as \( n \to \infty \).

Since the numerator is bounded and the denominator grows exponentially, the fraction \( \frac{\sin n}{2^n} \) approaches zero as \( n \to \infty \).

Conclude that the limit of the sequence \( \left\{ \frac{\sin n}{2^n} \right\} \) is zero.

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Key Concepts

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

Sequences and Limits

A sequence is an ordered list of numbers defined by a specific rule. The limit of a sequence is the value that the terms approach as the index goes to infinity. Understanding how to determine if a sequence converges (has a limit) or diverges (does not have a limit) is fundamental in calculus.

Behavior of Exponential Functions

Exponential functions like 2ⁿ grow very rapidly as n increases. When a sequence has terms divided by an exponential function with base greater than 1, the denominator grows faster than the numerator, often causing the sequence to approach zero. This property helps in evaluating limits involving exponential terms.

Boundedness of the Sine Function

The sine function oscillates between -1 and 1 for all real numbers. This boundedness means that sin(n) remains within fixed limits regardless of n. When combined with a rapidly growing denominator, the bounded numerator ensures the sequence terms become very small, aiding in limit evaluation.

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Graph of Sine and Cosine Function

Related Practice

Textbook Question

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) (−1)ᵏ · k / (2k + 1)

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Textbook Question

84–87. {Use of Tech} Sequences by recurrence relations

The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.

a.Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.

b.Use analytical methods to find the limit of the sequence.

{Use of Tech}aₙ₊₁ = √(2 + aₙ);a₀ = 3

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Textbook Question

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 2 to ∞) (−1)ᵏ (1 + 1/k)

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Textbook Question

Series of squares Prove that if ∑aₖ is a convergent series of positive terms, then the series ∑aₖ² also converges.

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Textbook Question

46–53. Decimal expansions

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).

53.0.00952̅ = 0.00952952…

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Textbook Question

48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.

2 / 4² + 2 / 5² + 2 / 6² + ⋯

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55–70. More sequencesFind the limit of the following sequences or determine that the sequence diverges.{sinn / 2ⁿ}

Verified video answer for a similar problem:

Key Concepts

Sequences and Limits

Behavior of Exponential Functions

Boundedness of the Sine Function

55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.

{sinn / 2ⁿ}