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 10.8.31
Capitolo 10, Problema 10.8.31

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from j = 1 to ∞) 5 / (j² + 4)

Guida verificata passo dopo passo
1
Identify the given series: \( \sum_{j=1}^{\infty} \frac{5}{j^{2} + 4} \). We want to determine if this series converges or diverges.
Recognize that the terms \( \frac{5}{j^{2} + 4} \) are positive and decrease as \( j \) increases, since the denominator grows quadratically.
Compare the given series to a known benchmark series. Notice that \( \frac{5}{j^{2} + 4} < \frac{5}{j^{2}} \) for all \( j \geq 1 \). The series \( \sum_{j=1}^{\infty} \frac{5}{j^{2}} \) is a constant multiple of the p-series \( \sum \frac{1}{j^{2}} \) with \( p = 2 > 1 \), which is known to converge.
Apply the Comparison Test: since \( \sum \frac{5}{j^{2}} \) converges and \( \frac{5}{j^{2} + 4} \leq \frac{5}{j^{2}} \), the original series \( \sum \frac{5}{j^{2} + 4} \) also converges.
Conclude that the series converges by the Comparison Test, justifying the answer based on the behavior of the terms and the known convergence of the p-series.

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:
4m

Concetti chiave

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

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates.
Video consigliato:
Percorso guidato
06:52
Convergence of an Infinite Series

Comparison Test

The Comparison Test involves comparing a given series to a second series whose convergence behavior is known. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges.
Video consigliato:
Percorso guidato
09:25
Direct Comparison Test

p-Series Test

A p-series is of the form ∑ 1/n^p, which converges if p > 1 and diverges otherwise. Recognizing that the given series resembles a p-series helps in applying this test to determine convergence by comparing the denominator's growth rate.
Video consigliato:
Percorso guidato
04:30
P-Series and Harmonic Series
Pratica correlata
Domanda del libro di testo

13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁. 

aₙ = (−1)ⁿ / 2ⁿ

47
views
Domanda del libro di testo

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 3 to ∞) 1 / lnk

36
views
Domanda del libro di testo

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 1 to ∞) k^(1/k)

47
views
Domanda del libro di testo

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


61. ∑ (k = 1 to ∞) ln((k + 1) / k)

75
views
Domanda del libro di testo

Why does the value of a converging alternating series with terms that are nonincreasing in magnitude lie between any two consecutive terms of its sequence of partial sums?

88
views
Domanda del libro di testo

17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.

∑ (k = 1 to ∞) 1 / (∛(5k + 3))

85
views