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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 10.2.59
Capitolo 10, Problema 10.2.59

Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
∑ (from n = 0 to ∞) e^(−2n)

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1
Identify the type of series given: \( \sum_{n=0}^{\infty} e^{-2n} \). Notice that this is a geometric series because each term can be written as \( (e^{-2})^n \).
Recall the formula for the sum of an infinite geometric series: if \( |r| < 1 \), then \( \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \), where \( a \) is the first term and \( r \) is the common ratio.
Determine the first term \( a \) and the common ratio \( r \) for this series. Here, \( a = e^{-2 \cdot 0} = 1 \) and \( r = e^{-2} \).
Check the convergence condition by verifying if \( |r| < 1 \). Since \( e^{-2} \) is a positive number less than 1, the series converges.
Use the sum formula to express the sum of the series as \( \frac{1}{1 - e^{-2}} \). This represents the sum of the infinite series.

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Geometric Series

A geometric series is a series where each term is obtained by multiplying the previous term by a constant ratio r. It converges if the absolute value of r is less than 1, and its sum can be found using the formula S = a / (1 - r), where a is the first term.
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Geometric Series

Convergence Criteria for Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. For geometric series, this depends on the common ratio, while for other series, tests like the comparison or ratio test may be used to determine convergence or divergence.
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Convergence of an Infinite Series

Exponential Functions in Series

Exponential functions of the form e^(−kn) decrease rapidly as n increases when k > 0. When used as terms in a series, they often form geometric series with ratio e^(−k), which helps in analyzing convergence and calculating sums.
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Exponential Functions
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Determining Convergence or Divergence

Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

∑ (from n=1 to ∞) (1 + 1/n)ⁿ

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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⁺¹ (ⁿ√10)]

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Domanda del libro di testo

Does the series

∑ (from n=1 to ∞) (1/n − 1/n²)

converge or diverge? Justify your answer.

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Domanda del libro di testo

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 0 to ∞) [((n + 1) / (n + 2))ⁿ]

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Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 1 to ∞) [ (x − 1)ⁿ / √n ]

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Suppose that aₙ > 0 and limₙ→∞ n²aₙ = 0. Prove that ∑aₙ converges.

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