Skip to main content
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.89b
Capitolo 10, Problema 10.R.89b

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

Guida verificata passo dopo passo
1
Identify the series given: \( \sum_{k=1}^{\infty} \frac{1}{k^5} \). This is a p-series with \( p = 5 \), which converges because \( p > 1 \).
Recall that the remainder \( R_n \) after \( n \) terms is the difference between the infinite sum and the partial sum up to \( n \): \( R_n = S - S_n \). We want to find an upper bound for \( R_n \) when \( n = 5 \).
Use the integral test remainder estimate for a decreasing positive function \( f(k) = \frac{1}{k^5} \). The upper bound for the remainder \( R_n \) is given by the integral from \( n \) to infinity of \( f(x) \): \[ R_n \leq \int_n^{\infty} \frac{1}{x^5} \, dx \]
Evaluate the improper integral: \[ \int_n^{\infty} \frac{1}{x^5} \, dx = \lim_{t \to \infty} \int_n^t x^{-5} \, dx \] Find the antiderivative of \( x^{-5} \), which is \( \frac{x^{-4}}{-4} = -\frac{1}{4x^4} \).
Apply the limits to the antiderivative: \[ \lim_{t \to \infty} \left( -\frac{1}{4t^4} + \frac{1}{4n^4} \right) = 0 + \frac{1}{4n^4} = \frac{1}{4n^4} \] Thus, the upper bound for the remainder \( R_n \) when \( n = 5 \) is \( \frac{1}{4 \times 5^4} \).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Convergent Series

A convergent series is an infinite sum whose partial sums approach a finite limit. For example, the series ∑ 1/k⁵ converges because its terms decrease rapidly and resemble a p-series with p > 1. Understanding convergence ensures that the remainder after n terms is meaningful and finite.
Video consigliato:
Percorso guidato
06:52
Convergence of an Infinite Series

Remainder (Error) in a Series Approximation

The remainder Rₙ is the difference between the infinite series sum and the partial sum up to n terms. It measures the error when approximating the series by a finite sum. Finding an upper bound for Rₙ helps estimate how close the partial sum is to the actual sum.
Video consigliato:
Percorso guidato
06:32
Alternating Series Remainder

Integral Test and Remainder Bounds

The integral test can be used to estimate the remainder of a decreasing positive term series. It states that the remainder Rₙ is less than or equal to the integral of the function from n to infinity. This provides a practical way to find upper bounds for the error in series approximations.
Video consigliato:
Percorso guidato
07:25
Integral Test
Pratica correlata
Domanda del libro di testo

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

∑ (from k = 0 to ∞)(tan⁻¹(k + 2) − tan⁻¹k)

81
views
Domanda del libro di testo

Sequences versus series

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

46
views
Domanda del libro di testo

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.


a.Find the first five terms a₀, a₁, ..., a₄ of each sequence.


25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80

32
views
Domanda del libro di testo

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)

41
views
Domanda del libro di testo

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

104
views
Domanda del libro di testo

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

81
views