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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 10.4.61
Chapter 10, Problem 10.4.61

48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
2 / 4² + 2 / 5² + 2 / 6² + ⋯

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1
Identify the series given: it is \( \frac{2}{4^2} + \frac{2}{5^2} + \frac{2}{6^2} + \cdots \). This is an infinite series where the general term can be written as \( a_n = \frac{2}{n^2} \) starting from \( n=4 \).
Recognize the type of series: since the terms involve \( \frac{1}{n^2} \), this resembles a p-series, which has the general form \( \sum \frac{1}{n^p} \). Here, \( p = 2 \).
Recall the p-series test: a p-series \( \sum \frac{1}{n^p} \) converges if and only if \( p > 1 \). Since \( p = 2 > 1 \), the series \( \sum \frac{1}{n^2} \) converges.
Since the series has a constant multiple of 2, factor it out: \( \sum_{n=4}^\infty \frac{2}{n^2} = 2 \sum_{n=4}^\infty \frac{1}{n^2} \). Multiplying a convergent series by a constant does not affect convergence.
Conclude that the given series converges by the p-series test because it is a constant multiple of a convergent p-series with \( p=2 \).

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Key Concepts

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

Convergence and Divergence of Series

A series converges if the sum of its terms approaches a finite limit as the number of terms increases indefinitely. If the sum grows without bound or oscillates, the series diverges. Understanding this distinction is fundamental to analyzing infinite series.
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Convergence of an Infinite Series

p-Series Test

The p-series test determines convergence based on the form ∑ 1/n^p. If p > 1, the series converges; if p ≤ 1, it diverges. This test is useful for series with terms involving powers of n, like the given series with terms 2/n².
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P-Series and Harmonic Series

Comparison Test

The comparison test involves comparing a given series to a known benchmark series to determine convergence or divergence. 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
Related Practice
Textbook Question

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

aₙ = (2ⁿ⁺¹) / (2ⁿ + 1)

Textbook Question

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

∑ (from k = 1 to ∞) (−1)ᵏ / √(k³ᐟ² + k)

Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) (2k⁴ + k) / (4k⁴ − 8k)

Textbook Question

55–70. More sequences

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


{tan⁻¹n / n}

1
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Textbook Question

13–52. Limits of sequences

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


{(1 + (2 / n))ⁿ}

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Textbook Question

1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.

∑ (from k = 1 to ∞) (k² / (k⁴ + k³ + 1))