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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 10.4.61
Capitolo 10, Problema 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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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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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
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45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) (−1)ᵏ · k / (2k + 1)

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55–70. More sequences

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


{sinn / 2ⁿ}

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84–87. {Use of Tech} Sequences by recurrence relations

The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.


a.Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.

b.Use analytical methods to find the limit of the sequence.



{Use of Tech}aₙ₊₁ = √(2 + aₙ);a₀ = 3

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Domanda del libro di testo

Series of squares Prove that if ∑aₖ is a convergent series of positive terms, then the series ∑aₖ² also converges.

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46–53. Decimal expansions

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).

53.0.00952̅ = 0.00952952…

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Growth rates of sequences

Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.


{n¹⁰⁰⁰ / 2ⁿ}

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