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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 10.3.57
Chapter 10, Problem 10.3.57

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.


57. ∑ (k = 1 to ∞) 1 / ((k + 6)(k + 7))

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1
Start by expressing the general term of the series: \( a_k = \frac{1}{(k+6)(k+7)} \). Our goal is to rewrite this term in a form that allows telescoping.
Use partial fraction decomposition to break \( a_k \) into simpler fractions. Set \( \frac{1}{(k+6)(k+7)} = \frac{A}{k+6} + \frac{B}{k+7} \) and solve for constants \( A \) and \( B \).
After finding \( A \) and \( B \), rewrite \( a_k \) as \( \frac{A}{k+6} + \frac{B}{k+7} \). This will help identify terms that cancel out when summing.
Write the partial sum \( S_n = \sum_{k=1}^n a_k \) by substituting the decomposed form of \( a_k \). Observe how most terms cancel out (telescope), leaving only a few terms from the beginning and end.
Express \( S_n \) explicitly in terms of \( n \), then evaluate the limit \( \lim_{n \to \infty} S_n \) to determine whether the series converges and find its sum if it does.

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Key Concepts

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

Telescoping Series

A telescoping series is a series whose partial sums simplify by canceling intermediate terms, leaving only a few terms from the beginning and end. This simplification makes it easier to find a formula for the nth partial sum and analyze convergence.
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Geometric Series

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into simpler fractions. For series like 1/((k+6)(k+7)), this technique helps rewrite terms so that the series telescopes, enabling easier summation and limit evaluation.
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Partial Fraction Decomposition: Distinct Linear Factors

Limit of Partial Sums and Convergence

The sum of an infinite series is the limit of its partial sums as n approaches infinity. If this limit exists and is finite, the series converges; otherwise, it diverges. Evaluating this limit determines the series' value or divergence.
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Related Practice
Textbook Question

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


23.∑ (k = 0 to ∞) (–9/10)ᵏ

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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 ∞) ((-1)ᵏ) / (k!)

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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).


49.0.037̅ = 0.037037…

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Textbook Question

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

∑ (from k = 1 to ∞) (10ᵏ + 1) / k¹⁰

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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 k (2ᵏ⁺¹ / (9ᵏ − 1))

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Textbook Question

1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.

∑ (from k = 1 to ∞) ((−1)ᵏ⁺¹) / (√2ᵏ + lnk)

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