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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 10.3.67
Chapter 10, Problem 10.3.67

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.


67. ∑ (k = 1 to ∞) 3 / (k² + 5k + 4)

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Start by factoring the denominator of the general term in the series. The denominator is \(k^2 + 5k + 4\). Factor it as \((k + 1)(k + 4)\).
Rewrite the general term \(\frac{3}{k^2 + 5k + 4}\) as \(\frac{3}{(k + 1)(k + 4)}\). Next, use partial fraction decomposition to express this as \(\frac{A}{k + 1} + \frac{B}{k + 4}\) for constants \(A\) and \(B\).
Set up the equation \(\frac{3}{(k + 1)(k + 4)} = \frac{A}{k + 1} + \frac{B}{k + 4}\). Multiply both sides by \((k + 1)(k + 4)\) to get \(3 = A(k + 4) + B(k + 1)\). Solve for \(A\) and \(B\) by choosing convenient values for \(k\) or by equating coefficients.
Once you find \(A\) and \(B\), write the nth partial sum \(S_n = \sum_{k=1}^n \frac{3}{(k + 1)(k + 4)} = \sum_{k=1}^n \left( \frac{A}{k + 1} + \frac{B}{k + 4} \right)\). Separate the sums and write out the terms to identify the telescoping pattern.
Use the telescoping property to cancel terms and find a simplified expression for \(S_n\). Then, evaluate \(\lim_{n \to \infty} S_n\) by analyzing the behavior of the remaining terms as \(n\) approaches infinity to determine if 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 evaluate the series' limit.
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Geometric Series

Partial Fraction Decomposition

Partial fraction decomposition breaks a rational function into simpler fractions that are easier to sum or integrate. For series like ∑ 3/(k² + 5k + 4), factoring the denominator and decomposing helps express terms in a telescoping form.
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Partial Fraction Decomposition: Distinct Linear Factors

Limit of Partial Sums and Convergence

The limit of the sequence of partial sums determines whether an infinite series converges or diverges. If the limit exists and is finite, the series converges to that value; otherwise, it diverges.
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Related Practice
Textbook Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.


{1 + cos(1⁄n)}

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Textbook Question

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 3 to ∞) 1 / (k − 2)⁴

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 ∞) 1 / (k² + 4)

Textbook Question

6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.


{1.00001ⁿ}

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Textbook Question

Compare the growth rates of {n¹⁰⁰} and {eⁿ⁄¹⁰⁰} as n → ∞.

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Textbook Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.


{(1 / n)¹⁄ⁿ}

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