Skip to main content
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 + ⋯

Verified step by step guidance
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 \( p=2 \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

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.
Recommended video:
06:52
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.
Recommended video:
04:30
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.
Recommended video:
09:25
Direct Comparison Test
Related Practice
Textbook Question

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)

75
views
Textbook Question

55–70. More sequences

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


{sinn / 2ⁿ}

52
views
Textbook Question

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

45
views
Textbook Question

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

71
views
Textbook Question

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…

41
views
Textbook Question

Growth rates of sequences

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


{n¹⁰⁰⁰ / 2ⁿ}

100
views