Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.6.7

Chapter 10, Problem 10.6.7

Is it possible for a series of positive terms to converge conditionally? Explain.

Verified step by step guidance

Recall the definitions: A series \( \sum a_n \) converges absolutely if \( \sum |a_n| \) converges, and it converges conditionally if \( \sum a_n \) converges but \( \sum |a_n| \) diverges.

Note that for a series of positive terms, each term \( a_n > 0 \), so \( |a_n| = a_n \). This means the series and its absolute value series are the same.

Since the series and its absolute value series are identical for positive terms, if the series converges, it must converge absolutely.

Therefore, a series with positive terms cannot converge conditionally because conditional convergence requires the absolute value series to diverge.

In summary, conditional convergence is only possible when the series has terms that are not all positive (i.e., some terms are negative or alternate in sign).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Positive Term Series

A series with positive terms means each term is greater than zero. Such series are always non-negative and cannot have terms that cancel each other out, which affects the type of convergence they can exhibit.

Absolute Convergence

A series converges absolutely if the series of the absolute values of its terms converges. For positive term series, absolute convergence is equivalent to regular convergence since all terms are positive.

Conditional Convergence

Conditional convergence occurs when a series converges, but the series of absolute values diverges. This typically requires terms of varying signs, so a series with only positive terms cannot converge conditionally.

Recommended video:

07:51

Choosing a Convergence Test

Related Practice

Textbook Question

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.

∑ (k = 2 to ∞) √k / (ln¹⁰ k)

Textbook Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

{Use of Tech} ∑ (k = 1 to ∞) 1 / (4lnk)

Textbook Question

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

∑ (from k = 1 to ∞)1 / ln(eᵏ + 1)

Textbook Question

21–42. Geometric series Evaluate each geometric series or state that it diverges.

39.∑ (k = 2 to ∞) (–0.15)ᵏ

Textbook Question

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) (2ᵏ / k⁹⁹)

Textbook Question

35–44. Limits of sequences Write the terms a₁, a₂, a₃, and a₄ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why.

aₙ = 3 + cos(π*ⁿ) ; n = 1, 2, 3, …