Domanda del libro di testo
Finding Taylor Polynomials
In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.
f(x) = √(1 − x),a = 0
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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 10.5.18
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 ∞) [(-1)ⁿ n² e⁻ⁿ]
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Identify the given series: \( \sum_{n=1}^{\infty} (-1)^n n^2 e^{-n} \). This is an alternating series because of the factor \( (-1)^n \), which causes the terms to alternate in sign.
Check the absolute value of the terms: \( a_n = n^2 e^{-n} \). Since \( e^{-n} = \frac{1}{e^n} \), the terms involve a polynomial factor \( n^2 \) multiplied by an exponential decay \( e^{-n} \).
Determine if the terms \( a_n = n^2 e^{-n} \) approach zero as \( n \to \infty \). Because exponential decay dominates polynomial growth, \( a_n \to 0 \). This is a necessary condition for convergence.
Consider the absolute convergence by examining the series \( \sum_{n=1}^{\infty} n^2 e^{-n} \). Since \( e^{-n} \) decays faster than any polynomial grows, this series converges by comparison to a convergent geometric series.
Conclude that since the series converges absolutely (the series of absolute values converges), the original alternating series \( \sum_{n=1}^{\infty} (-1)^n n^2 e^{-n} \) converges as well.
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Convergence and Divergence of Infinite Series
An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely; otherwise, it diverges. Understanding this concept is fundamental to analyzing whether a given series sums to a finite value or not.
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Percorso guidato
06:52
Convergence of an Infinite Series
Alternating Series Test
The Alternating Series Test determines convergence for series whose terms alternate in sign. If the absolute value of the terms decreases monotonically to zero, the series converges. This test is useful for series with factors like (-1)^n.
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Percorso guidato
10:54Alternating Series Test
Comparison and Limit Comparison Tests
These tests compare a given series to a known benchmark series to determine convergence or divergence. By analyzing the behavior of terms relative to simpler series, one can infer the original series' behavior, especially when exponential or polynomial terms are involved.
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Percorso guidato
07:45Limit Comparison Test
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Domanda del libro di testo
Determining Convergence or Divergence
In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]
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Domanda del libro di testo
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = 2 + (0.1)ⁿ
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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 / (n³ 3ⁿ) ]
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Domanda del libro di testo
In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ n! xⁿ / nⁿ ]
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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 = 0 to ∞) [ n xⁿ / (4ⁿ (n² + 1)) ]
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