Domanda del libro di testo
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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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 10, Problema 10.6.66
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]
Guida verificata passo dopo passo1
Identify the general term of the series: \(a_n = \left( \frac{n+1}{n+2} \right)^n\).
To determine convergence or divergence, consider using the Root Test or the Ratio Test. The Root Test is often simpler for terms raised to the power \(n\).
Apply the Root Test by evaluating \(\lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \left| \frac{n+1}{n+2} \right|\).
Simplify the limit: since \(\frac{n+1}{n+2} = 1 - \frac{1}{n+2}\), the limit approaches 1 as \(n\) approaches infinity.
Interpret the Root Test result: if the limit equals 1, the test is inconclusive, so consider using the Ratio Test or analyze the behavior of \(a_n\) directly to decide on convergence or divergence.
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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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Percorso guidato
06:52
Convergence of an Infinite Series
Limit of the General Term
A necessary condition for a series to converge is that its general term approaches zero as n approaches infinity. If the limit of the term does not equal zero, the series diverges. Evaluating this limit helps quickly assess the possibility of convergence.
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05:50One-Sided Limits
Root Test for Series Convergence
The root test uses the nth root of the absolute value of the general term to determine convergence. If the limit of this root is less than one, the series converges absolutely; if greater than one, it diverges. This test is particularly useful for series with terms raised to the nth power.
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07:15Root Test
Pratica correlata
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
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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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 ∞) (n / (3n + 1))ⁿ
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Domanda del libro di testo
Find the first four nonzero terms of the Taylor series for the functions in Exercises 1–10.
6. (1 - x/3)^4
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Domanda del libro di testo
Suppose that aₙ > 0 and limₙ→∞ n²aₙ = 0. Prove that ∑aₙ converges.
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