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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.R.25b
Capitolo 10, Problema 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

Guida verificata passo dopo passo
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.

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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.
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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.
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Finding Limits by Direct Substitution
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