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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 10.6.80
Capitolo 10, Problema 10.6.80

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 0 to ∞) [(2n + 3)(2ⁿ + 3) / (3ⁿ + 2)]

Guida verificata passo dopo passo
1
First, write down the general term of the series: \(a_n = \frac{(2n + 3)(2^n + 3)}{3^n + 2}\).
To determine convergence or divergence, analyze the behavior of \(a_n\) as \(n\) approaches infinity. Focus on the dominant terms in the numerator and denominator for large \(n\).
Approximate the dominant terms: in the numerator, \((2n + 3)(2^n + 3) \approx 2n \cdot 2^n = 2n \cdot 2^n\), and in the denominator, \(3^n + 2 \approx 3^n\) for large \(n\).
Simplify the approximate form of \(a_n\) for large \(n\): \(a_n \approx \frac{2n \cdot 2^n}{3^n} = 2n \left(\frac{2}{3}\right)^n\).
Use the Divergence Test (also called the nth-term test): if \(\lim_{n \to \infty} a_n \neq 0\), the series diverges. Otherwise, consider applying a convergence test such as the Ratio Test or Root Test to confirm convergence.

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Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining whether a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of the given series.
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Convergence of an Infinite Series

Comparison and Limit Comparison Tests

These tests help determine convergence by comparing the given series to a known benchmark series. The Comparison Test involves directly comparing terms, while the Limit Comparison Test uses the limit of the ratio of terms. They are useful when series involve exponential or polynomial expressions.
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Limit Comparison Test

Ratio Test

The Ratio Test examines the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges absolutely; if greater than one, it diverges. This test is particularly effective for series with exponential terms like powers of n.
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Ratio Test
Pratica correlata
Domanda del libro di testo

Use the Integral Test to show that the series

∑ (from n=0 to ∞) e^(−n²)

converges.

Domanda del libro di testo

Determining Convergence or Divergence

In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑(from n=1 to ∞) [(2 + (−1)ⁿ) / 1.25ⁿ]

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 = 2 to ∞) [(3n)! / (n!)³]

Domanda del libro di testo

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) (1 + cos n) / n²

Domanda del libro di testo

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) 1 / (1 + 2² + 3² + ⋯ + n²)

Domanda del libro di testo

Find the value of b for which

1 + eᵇ + e²ᵇ + e³ᵇ + ⋯ = 9.

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