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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 10.4.51
Chapter 10, Problem 10.4.51

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

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1
First, recognize the series given: \( \sum_{k=1}^{\infty} \frac{1}{(3k + 1)(3k + 4)} \). This is an infinite series with positive terms, so we can consider tests for convergence of positive term series.
Next, try to simplify the general term using partial fraction decomposition. Express \( \frac{1}{(3k + 1)(3k + 4)} \) as \( \frac{A}{3k + 1} + \frac{B}{3k + 4} \) and solve for constants \( A \) and \( B \).
After finding \( A \) and \( B \), rewrite the series as \( \sum_{k=1}^{\infty} \left( \frac{A}{3k + 1} + \frac{B}{3k + 4} \right) \). This often leads to a telescoping series where many terms cancel out.
Identify the telescoping pattern by writing out the first few terms explicitly and observe how terms cancel when summed.
Finally, use the telescoping property to find the partial sums and analyze their limit as \( n \to \infty \). If the limit exists and is finite, the series converges; otherwise, it diverges.

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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 Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely. Otherwise, it diverges. Understanding this concept is fundamental to analyzing whether a given series sums to a finite value or not.
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Convergence of an Infinite Series

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into simpler fractions that are easier to sum or analyze. For series with terms like 1/((3k+1)(3k+4)), this technique helps rewrite terms to identify telescoping behavior or apply known convergence tests.
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Partial Fraction Decomposition: Distinct Linear Factors

Telescoping Series Test

A telescoping series is one where many terms cancel out when the partial sums are expanded, simplifying the limit evaluation. Recognizing telescoping patterns after decomposition allows for straightforward determination of convergence by examining the remaining terms.
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Alternating Series Test
Related Practice
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 ∞)k! / (kᵏ + 3)

Textbook Question

21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.

aₙ₊₁ = 2aₙ; a₁ = 2

Textbook Question

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


∑ (k = 1 to ∞) 20 / (∛k + √k)

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.

1 / 1! + 4 / 2! + 9 / 3! + 16 / 4! + ⋯

Textbook Question

For what values of p does the series ∑ (k = 10 to ∞) 1 / kᵖ converge (initial index is 10)? For what values of p does it diverge?