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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 10.8.31
Chapter 10, Problem 10.8.31

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

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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.

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Key Concepts

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

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.
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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.
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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.
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P-Series and Harmonic Series
Related Practice
Textbook Question

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

aₙ = (−1)ⁿ / 2ⁿ

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Textbook Question

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

∑ (k = 3 to ∞) 1 / lnk

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Textbook Question

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)

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Textbook Question

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)

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Textbook Question

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?

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Textbook Question

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))

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