Domanda del libro di testo
In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ ( n / (n + 1) )ⁿ^ ² ] xⁿ (Hint: Apply the Root Test.)
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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 10.8.25
Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = x³ − 2x + 4,a = 2
Guida verificata passo dopo passo1
Identify the function and the point about which the Taylor series is generated: here, the function is \(f(x) = x^{3} - 2x + 4\) and the center is \(a = 2\).
Recall the formula for the Taylor series of a function \(f\) about \(x = a\):
\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^{n}\]
where \(f^{(n)}(a)\) is the \(n\)-th derivative of \(f\) evaluated at \(x = a\).
Compute the derivatives of \(f(x)\) up to the order needed (usually until the derivatives become zero or a pattern emerges):
- \(f(x) = x^{3} - 2x + 4\)
- \(f'(x) = 3x^{2} - 2\)
- \(f''(x) = 6x\)
- \(f^{(3)}(x) = 6\)
- \(f^{(4)}(x) = 0\) (and all higher derivatives are zero).
Evaluate each derivative at \(x = 2\):
- \(f(2) = 2^{3} - 2(2) + 4\)
- \(f'(2) = 3(2)^{2} - 2\)
- \(f''(2) = 6(2)\)
- \(f^{(3)}(2) = 6\)
- \(f^{(4)}(2) = 0\).
Write the Taylor series expansion using the formula and the evaluated derivatives:
\[f(x) = f(2) + \frac{f'(2)}{1!}(x - 2) + \frac{f''(2)}{2!}(x - 2)^{2} + \frac{f^{(3)}(2)}{3!}(x - 2)^{3} + \cdots\]
Since higher derivatives are zero, the series will terminate here.
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Taylor Series
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point a. It approximates the function near that point using polynomial terms, where each term involves the nth derivative evaluated at a, multiplied by (x - a)^n and divided by n!.
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08:42
Taylor Series
Maclaurin Series
The Maclaurin series is a special case of the Taylor series centered at a = 0. It expresses a function as an infinite sum of derivatives evaluated at zero, useful for approximating functions near zero. Understanding this helps distinguish between general Taylor expansions and those specifically at zero.
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08:26Convergence of Taylor & Maclaurin Series
Derivatives and Their Role in Series Expansion
Derivatives of a function at the point a determine the coefficients of the Taylor series. Calculating successive derivatives and evaluating them at a provides the necessary values to build each term of the series, reflecting the function's behavior near a.
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Percorso guidato
06:00Geometric Series
Pratica correlata
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 = 1 to ∞) (1 − 1/n)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 = 0 to ∞) (x + 5)ⁿ
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Domanda del libro di testo
Is it true that a sequence {aₙ} of positive numbers must converge if it is bounded above? Give reasons for your answer.
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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) = sin x,a = 0
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Domanda del libro di testo
Finding Taylor Series
Use substitution (as in Formula (7)) to find the Taylor series at x = 0 of the functions in Exercises 1–12.
e⁻ˣ/²
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