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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 10.3.59
Chapter 10, Problem 10.3.59

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))

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Start by expressing the general term of the series: \( a_k = \frac{4}{(4k - 3)(4k + 1)} \). Our goal is to rewrite this term in a form that reveals telescoping behavior.
Use partial fraction decomposition to break \( a_k \) into simpler fractions. Assume \( \frac{4}{(4k - 3)(4k + 1)} = \frac{A}{4k - 3} + \frac{B}{4k + 1} \) and solve for constants \( A \) and \( B \).
Once you find \( A \) and \( B \), rewrite \( a_k \) as the difference of two fractions, which will help terms cancel out when summing from \( k = -3 \) to \( n \).
Write the partial sum \( S_n = \sum_{k=-3}^n a_k \) using the decomposed form. Observe how most terms cancel out due to the telescoping nature, leaving only a few terms from the start and end of the sum.
Express \( S_n \) explicitly in terms of \( n \), then analyze \( \lim_{n \to \infty} S_n \) to determine whether the series converges or diverges.

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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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Partial Sums and Their Formulas

The partial sum Sₙ of a series is the sum of its first n terms. Finding a closed-form expression for Sₙ helps in evaluating the behavior of the series as n approaches infinity, which is essential for determining convergence or divergence.
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Limit of a Sequence

The limit of the sequence of partial sums, limₙ→∞ Sₙ, determines the sum of an infinite series if it exists. If this limit is finite, the series converges; otherwise, it diverges. Understanding limits is crucial for evaluating infinite series.
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Related Practice
Textbook Question

43–44. Periodic doses

Suppose you take a dose of m mg of a particular medication once per day. Assume f equals the fraction of the medication that remains in your blood one day later. Just after taking another dose of medication on the second day, the amount of medication in your blood equals the sum of the second dose and the fraction of the first dose remaining in your blood, which is m + mf. Continuing in this fashion, the amount of medication in your blood just after your nth dose is


Aₙ = m + mf + ⋯ + mfⁿ⁻¹.


For the given values of f and m, calculate A₅, A₁₀, A₃₀, and lim (n → ∞) Aₙ. Interpret the meaning of the limit lim (n → ∞) Aₙ.


43.f = 0.25,m = 200 mg

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!)

Textbook Question

38–39. Examining a series two ways Determine whether the following series converge using either the Comparison Test or the Limit Comparison Test. Then use another method to check your answer.

39. ∑ (k = 1 to ∞) 1 / (k² + 2k + 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 = 8ⁿ / n!

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Textbook Question

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)