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.R.81
Chapter 10, Problem 10.R.81

77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−1)ᵏ⁺¹(k² + 4) / (2k² + 1)

Verified step by step guidance
1
Identify the given series: \( \sum_{k=1}^{\infty} (-1)^{k+1} \frac{k^2 + 4}{2k^2 + 1} \). This is an alternating series because of the factor \( (-1)^{k+1} \).
Check for absolute convergence by considering the series of absolute values: \( \sum_{k=1}^{\infty} \left| (-1)^{k+1} \frac{k^2 + 4}{2k^2 + 1} \right| = \sum_{k=1}^{\infty} \frac{k^2 + 4}{2k^2 + 1} \).
Analyze the behavior of the terms \( \frac{k^2 + 4}{2k^2 + 1} \) as \( k \to \infty \). Simplify the expression by dividing numerator and denominator by \( k^2 \) to find the limit of the terms.
Determine if the series of absolute values converges or diverges by comparing it to a known benchmark series, such as a \( p \)-series or using the Limit Comparison Test.
If the series of absolute values diverges, apply the Alternating Series Test to the original series by checking if the terms \( \frac{k^2 + 4}{2k^2 + 1} \) decrease monotonically to zero. Based on this, conclude whether the original series converges conditionally or diverges.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

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

Absolute Convergence

A series ∑a_k converges absolutely if the series of absolute values ∑|a_k| converges. Absolute convergence implies convergence regardless of the sign of terms, and it guarantees the sum is well-defined and stable under rearrangement.
Recommended video:
07:51
Choosing a Convergence Test

Conditional Convergence

A series converges conditionally if it converges, but does not converge absolutely. This means ∑a_k converges, but ∑|a_k| diverges. Conditional convergence often occurs in alternating series where the terms decrease in magnitude but the absolute series diverges.
Recommended video:
07:51
Choosing a Convergence Test

Alternating Series Test

The Alternating Series Test states that an alternating series ∑(−1)^k b_k converges if the sequence b_k is positive, decreasing, and approaches zero. This test helps determine conditional convergence when absolute convergence fails.
Recommended video:
10:54
Alternating Series Test
Related Practice
Textbook Question

77–87. Absolute or conditional convergence

Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞)(−1)ᵏk·e⁻ᵏ

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.


59. ∑ (k = –3 to ∞) 4 / ((4k – 3)(4k + 1))

Textbook Question

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

∑ (k = 2 to ∞) 1 / (klnk)

Textbook Question

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = (2ⁿ + 5ⁿ⁺¹) / 5ⁿ

1
views
Textbook Question

b.Does the series ∑ (from k = 1 to ∞) k/(k + 1) converge? Why or why not?

Textbook Question

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = 8ⁿ / n!

1
views