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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.R.9a
Chapter 10, Problem 10.R.9a

Sequences versus series
a.Find the limit of the sequence { (−⁴⁄₅)ᵏ }.

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1
Identify the given sequence as \( a_k = \left(-\frac{4}{5}\right)^k \), where \( k \) is a positive integer.
Recall that the limit of a sequence \( a_k \) as \( k \to \infty \) depends on the behavior of the base \( r = -\frac{4}{5} \) raised to the power \( k \).
Since \( |r| = \left| -\frac{4}{5} \right| = \frac{4}{5} < 1 \), the terms \( r^k \) approach zero as \( k \to \infty \).
Consider the effect of the negative sign: the terms alternate in sign but their magnitude decreases to zero, so the sequence converges to zero.
Therefore, the limit of the sequence \( \left\{ \left(-\frac{4}{5}\right)^k \right\} \) as \( k \to \infty \) 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 Their Limits

A sequence is an ordered list of numbers defined by a specific formula for its terms. The limit of a sequence is the value that the terms approach as the index goes to infinity. Understanding how to find limits helps determine the long-term behavior of sequences.
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Introduction to Sequences

Geometric Sequences

A geometric sequence is one where each term is found by multiplying the previous term by a constant ratio. The general form is a_k = a * r^k, where r is the common ratio. Recognizing geometric sequences simplifies finding limits and sums.
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Limit of a Geometric Sequence with |r| < 1

If the absolute value of the common ratio r in a geometric sequence is less than 1, the terms approach zero as k approaches infinity. This property is crucial for evaluating limits of sequences like (−4/5)^k, where |−4/5| = 0.8 < 1.
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Related Practice
Textbook Question

89–90. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, complete the following.


b. Find an upper bound for the remainder Rₙ.


89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5

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Textbook Question

27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.

∑ (from k = 0 to ∞)((1/3)ᵏ + (4/3)ᵏ)

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Textbook Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞)k⁵ e⁻ᵏ

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Textbook Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞)k! / (eᵏ kᵏ)

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Textbook Question

89–90. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, complete the following.


c. Find lower and upper bounds (Lₙ and Uₙ respectively) for the exact value of the series.


89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5

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Textbook Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 3 to ∞)ln(k) / k³ᐟ²

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