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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblema 10.PE.20
Capitolo 10, Problema 10.PE.20

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


∑ (from n = 2 to ∞) -2/[n(n+1)]

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1
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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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.
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Convergence of an Infinite Series

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.
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Percorso guidato
10:07
Partial Fraction Decomposition: Distinct Linear Factors

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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06:00
Geometric Series
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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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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)

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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 ∞) (csch n)xⁿ

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Determining Convergence of Sequences

Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.


aₙ = 1 + (0.9)ⁿ

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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]

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Maclaurin Series

Find Taylor series at x = 0 for the functions in Exercises 63–70.

cos (x³/√5)

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