Domanda del libro di testo
Convergence or Divergence
Which of the series in Exercises 57–64 converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ (n!)ⁿ] / [n^(n²)]
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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 10.9.45
Error Estimates
The approximation eˣ = 1 + x + (x² / 2) is used when x is small. Use the Remainder Estimation Theorem to estimate the error when |x| < 0.1.
Guida verificata passo dopo passo1
Identify the Taylor polynomial used for approximating \(e^x\). Here, the approximation is \(1 + x + \frac{x^2}{2}\), which corresponds to the Taylor polynomial of degree 2 centered at 0.
Recall the Remainder Estimation Theorem (Lagrange form of the remainder) for the Taylor polynomial of degree \(n\): the error \(R_n(x)\) satisfies
\(R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}\)
for some \(c\) between 0 and \(x\).
Since the polynomial is degree 2, the remainder term involves the third derivative of \(f(x) = e^x\). Note that all derivatives of \(e^x\) are \(e^x\), so
\(f^{(3)}(c) = e^c\).
To estimate the error when \(|x| < 0.1\), find an upper bound for \(|R_2(x)|\) by bounding \(|e^c|\) for \(c\) between 0 and \(x\). Since \(|x| < 0.1\), \(c\) lies in \([-0.1, 0.1]\), and \(e^c\) is maximized at \(c = 0.1\).
Write the error bound as
\(|R_2(x)| \leq \frac{e^{0.1}}{3!} |x|^3 = \frac{e^{0.1}}{6} |x|^3\).
This expression gives an estimate of the maximum error when using the given approximation for \(|x| < 0.1\).
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Taylor Polynomial Approximation
A Taylor polynomial approximates a function near a point using a finite sum of its derivatives at that point. For eˣ, the polynomial 1 + x + (x² / 2) is the second-degree Taylor polynomial centered at 0, providing a close estimate when x is near zero.
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07:00
Taylor Polynomials
Remainder Estimation Theorem (Lagrange Form)
This theorem gives a bound on the error between a function and its Taylor polynomial approximation. It states that the remainder term is bounded by the maximum value of the next derivative times |x|^(n+1) divided by (n+1)!, ensuring control over the approximation error.
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Percorso guidato
07:59Estimating the Area Under a Curve Using Left Endpoints
Bounding the Error for Small x
When |x| is small, higher powers of x become very small, reducing the error in the approximation. By evaluating the maximum of the next derivative on the interval and applying the remainder formula, one can estimate how close the polynomial is to the actual function value.
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Percorso guidato
04:57Determining Error and Relative Error
Pratica correlata
Domanda del libro di testo
In Exercises 53–56, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (n / (n² + 1))]
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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 ∞) 1 / (2√n + ³√n)
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Domanda del libro di testo
Using the Ratio Test
In Exercises 1–8, use the Ratio Test to determine whether each series converges absolutely or diverges.
∑(from n=2 to ∞) [(3ⁿ⁺²) / ln(n)]
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Domanda del libro di testo
Make up a geometric series ∑a rⁿ⁻¹ that converges to the number 5 if
b. a = 13/2
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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 = (xⁿ / (2n + 1))^(1/n),x > 0
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