Skip to main content
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.25b
Chapter 10, Problem 10.R.25b

25–26. Recursively defined sequences
The following sequences {aₙ} from n = 0 to ∞ are defined by a recurrence relation. Assume each sequence is monotonic and bounded.


b.Determine the limit of each sequence.


25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80

Verified step by step guidance
1
Identify the recurrence relation given: \(a_{n+1} = \frac{1}{2} a_n + 8\) with initial value \(a_0 = 80\).
Assuming the sequence converges to a limit \(L\), use the property that as \(n \to \infty\), \(a_n\) and \(a_{n+1}\) both approach \(L\).
Set up the equation for the limit by substituting \(a_n = L\) and \(a_{n+1} = L\) into the recurrence relation: \(L = \frac{1}{2} L + 8\).
Solve the resulting algebraic equation for \(L\) to find the limit of the sequence.
Verify that the sequence is monotonic and bounded to confirm that the limit found is valid.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Recursively Defined Sequences

A recursively defined sequence is one where each term is defined based on one or more previous terms using a recurrence relation. Understanding how to express terms in terms of earlier ones is essential for analyzing the sequence's behavior and finding limits.
Recommended video:
Guided course
6:40
Arithmetic Sequences - Recursive Formula

Monotonicity and Boundedness

A sequence is monotonic if it is either non-increasing or non-decreasing, and bounded if its terms lie within fixed upper and lower limits. These properties guarantee the existence of a limit for the sequence, which is crucial when determining convergence.

Finding Limits of Linear Recurrence Relations

For linear recurrence relations like aₙ₊₁ = r aₙ + c, the limit can be found by setting the limit L equal to its own recurrence: L = rL + c. Solving this equation gives the sequence's limit, assuming |r| < 1 to ensure convergence.
Recommended video:
05:21
Finding Limits by Direct Substitution
Related Practice
Textbook Question

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

∑ (from k = 1 to ∞)ln((2k + 1) / (2k − 1))

Textbook Question

Sequences versus series

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

1
views
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 ∞)3 / (2 + eᵏ)

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

Textbook Question

77–87. Absolute or conditional convergence

Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞)(−1)ᵏ⁺¹ / k³⁄⁷

Textbook Question

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

∑ (from j = 0 to ∞)2 ⋅ 4ʲ / (2j + 1)!