Domanda del libro di testo
Limits by power series Use Taylor series to evaluate the following limits.
lim ₙ → ₄ ln (x - 3)/(x² - 16)
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Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 11, Problema 11.R.49
Limits by power series Use Taylor series to evaluate the following limits.
lim ₙ → 0 (x²/2 - 1 + cos x)/x⁴
Guida verificata passo dopo passo1
Recall the Taylor series expansion of \( \cos x \) around \( x = 0 \):
\[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \]
Substitute the Taylor series expansion of \( \cos x \) into the given expression:
\[ \frac{\frac{x^2}{2} - 1 + \cos x}{x^4} = \frac{\frac{x^2}{2} - 1 + \left(1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \right)}{x^4} \]
Simplify the numerator by combining like terms:
\[ \frac{x^2}{2} - 1 + 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots = \frac{x^4}{24} - \cdots \]
Rewrite the expression with the simplified numerator:
\[ \frac{\frac{x^4}{24} - \cdots}{x^4} \]
Divide each term in the numerator by \( x^4 \) and then take the limit as \( x \to 0 \). The higher order terms vanish, leaving the constant term from the division.
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Taylor Series Expansion
A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. It approximates functions near that point, allowing complex expressions like cosine to be expressed as polynomials, which simplifies limit evaluation.
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Taylor Series
Limit Evaluation Using Series
By substituting the Taylor series into the limit expression, we can rewrite the limit in terms of powers of x. This method helps identify dominant terms and simplifies the limit calculation, especially when direct substitution leads to indeterminate forms.
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Percorso guidato
06:00Geometric Series
Handling Indeterminate Forms
Limits that result in forms like 0/0 require algebraic manipulation or series expansion to resolve. Using power series expansions transforms the expression into a form where the limit can be directly computed by canceling terms or evaluating the leading coefficients.
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6:20Circles in General Form
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Power series from the geometric series Use the geometric series a Σₖ ₌ ₀ ∞ (x)ᵏ = 1/(1 - x), for |x| < 1, to determine the Maclaurin series and the interval of convergence for the following functions.
ƒ(x) = 1/(1 - x²)
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sinh (-1)
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Approximating ln 2 Consider the following three ways to approximate
ln 2.
d. At what value of x should the series in part (c) be evaluated to approximate ln 2? Write the resulting infinite series for ln 2.
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Domanda del libro di testo
Power series from the geometric series Use the geometric series a Σₖ ₌ ₀ ∞ (x)ᵏ = 1/(1 - x), for |x| < 1, to determine the Maclaurin series and the interval of convergence for the following functions.
ƒ(x) = 1/(1 + 5x)
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Domanda del libro di testo
Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = sinh (-3x), n = 3, a = 0
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