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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 10.4.31
Capitolo 10, Problema 10.4.31

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 = 3 to ∞) 1 / (k − 2)⁴

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1
Identify the given series: \( \sum_{k=3}^{\infty} \frac{1}{(k-2)^4} \). Notice that the term inside the summation can be rewritten as \( \frac{1}{n^4} \) by letting \( n = k - 2 \). This shifts the index to start from \( n=1 \).
Recognize that the series is a p-series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) where \( p = 4 \). The p-series test states that such a series converges if and only if \( p > 1 \).
Since \( p = 4 > 1 \), the p-series test indicates that the series converges.
Optionally, you could apply the Integral Test by considering the function \( f(x) = \frac{1}{x^4} \) for \( x \geq 1 \), which is positive, continuous, and decreasing. Then evaluate the improper integral \( \int_1^{\infty} \frac{1}{x^4} \, dx \) to confirm convergence.
Conclude that by the p-series test (and optionally the Integral Test), the series \( \sum_{k=3}^{\infty} \frac{1}{(k-2)^4} \) converges.

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Divergence Test

The Divergence Test states that if the limit of the terms of a series does not approach zero, the series diverges. It is a quick initial check to determine if a series cannot converge, but if the limit is zero, the test is inconclusive.
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Percorso guidato
05:44
Divergence Test (nth Term Test)

Integral Test

The Integral Test relates a series to an improper integral by comparing the sum of terms to the integral of a corresponding function. If the integral converges, so does the series; if the integral diverges, the series diverges as well. It requires the function to be positive, continuous, and decreasing.
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Percorso guidato
07:25
Integral Test

p-series Test

A p-series is a series of the form ∑ 1/n^p. It converges if and only if p > 1 and diverges otherwise. This test is useful for quickly determining convergence of series with terms involving powers of n.
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Percorso guidato
04:30
P-Series and Harmonic Series
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Domanda del libro di testo

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.

1 / 1! + 4 / 2! + 9 / 3! + 16 / 4! + ⋯

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Domanda del libro di testo

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.


{1 + cos(1⁄n)}

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Domanda del libro di testo

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 1 to ∞) 1 / (k² + 4)

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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.


67. ∑ (k = 1 to ∞) 3 / (k² + 5k + 4)

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Domanda del libro di testo

6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.


{1.00001ⁿ}

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For what values of p does the series ∑ (k = 10 to ∞) 1 / kᵖ converge (initial index is 10)? For what values of p does it diverge?

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