Ch. 10 - Infinite Sequences and Series

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 10 - Infinite Sequences and Series

Problem 10.PE.20

Chapter 10, Problem 10.PE.20

Convergent Series
Find the sums of the series in Exercises 19–24.

∑ (from n = 2 to ∞) -2/[n(n+1)]

Verified step by step guidance

Recognize that the series is given by \( \sum_{n=2}^{\infty} \frac{-2}{n(n+1)} \). This is a rational function that can be decomposed using partial fractions.

Express the general term \( \frac{-2}{n(n+1)} \) as a difference of two simpler fractions: find constants \( A \) and \( B \) such that \( \frac{-2}{n(n+1)} = \frac{A}{n} + \frac{B}{n+1} \).

Solve for \( A \) and \( B \) by multiplying both sides by \( n(n+1) \) and equating coefficients, which will allow you to rewrite the series as a telescoping series.

Write out the first few terms of the series after partial fraction decomposition to observe the telescoping pattern, where many terms cancel out.

Use the telescoping property to find the sum of the infinite series by taking the limit as \( N \to \infty \) of the partial sums \( S_N \), which simplifies to a finite value.

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Key Concepts

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

Convergent Series

A series is convergent if the sum of its infinite terms approaches a finite limit. Determining convergence often involves tests or recognizing known series forms. For this problem, understanding convergence ensures the sum exists and can be calculated.

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into simpler fractions. This technique is useful for series like ∑ -2/[n(n+1)] because it allows rewriting terms to reveal telescoping behavior, simplifying the sum calculation.

Telescoping Series

A telescoping series is one where many terms cancel out when partial sums are expanded. Recognizing telescoping allows easy computation of the series sum by focusing on the first and last terms of the partial sums, avoiding complicated infinite sums.

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Textbook Question

Theory and Examples

Suppose that a₁, a₂, a₃, …, aₙ are positive numbers satisfying the following conditions:

i) a₁ ≥ a₂ ≥ a₃ ≥ …;

ii) the series a₂ + a₄ + a₈ + a₁₆ + … diverges.

Show that the series

a₁/1 + a₂/2 + a₃/3 + …

diverges.

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Textbook Question

Power Series

In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.

∑ (from n = 1 to ∞) (x + 4)ⁿ/(n3ⁿ)

Textbook Question

Absolute and Conditional Convergence

Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ (√(n² + n) − n)]

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Textbook Question

Determining Convergence of Series

Which of the series in Exercises 25–44 converge absolutely, which converge conditionally, and which diverge? Give reasons for your answers.

∑ (from n = 2 to ∞) 1/[n(ln n)²]

Textbook Question

30. b. By differentiating the series in part (a) term by term, show that