Domanda del libro di testo
Limits Evaluate the following limits using Taylor series.
lim ₓ→∞ x(e¹/ˣ − 1)
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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 11.3.41
Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.
(1 + x⁴)⁻¹
Guida verificata passo dopo passo1
Recall the Taylor series for the function \( \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots \), which is valid for \( |x| < 1 \).
Notice that the given function \( (1 + x^4)^{-1} \) can be rewritten as \( \frac{1}{1 - (-x^4)} \). This means we can use the geometric series formula by substituting \( -x^4 \) in place of \( x \).
Substitute \( -x^4 \) into the geometric series to get the Taylor series: \[ \frac{1}{1 + x^4} = \sum_{n=0}^{\infty} (-x^4)^n = \sum_{n=0}^{\infty} (-1)^n x^{4n} = 1 - x^4 + x^8 - x^{12} + \cdots \]
Identify the first four nonzero terms from the series expansion: these are the terms corresponding to \( n = 0, 1, 2, 3 \), which are \( 1, -x^4, x^8, -x^{12} \).
Write the final Taylor series approximation centered at 0 using these four terms: \[ 1 - x^4 + x^8 - x^{12} \]. This is the first four nonzero terms of the Taylor series for \( (1 + x^4)^{-1} \).
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Taylor Series Expansion
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point, usually centered at zero (Maclaurin series). It approximates functions using polynomials, making complex functions easier to analyze and manipulate.
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Taylor Series
Manipulating Known Series
Using known Taylor series expansions, such as for (1 - x)⁻¹, allows us to substitute expressions and find series for more complex functions. By replacing variables and adjusting terms, we can derive new series without computing derivatives from scratch.
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Percorso guidato
06:00Geometric Series
Identifying Nonzero Terms
When expanding a function into a Taylor series, it is important to recognize which terms are nonzero to write the first few significant terms. This involves understanding the powers of x that appear and simplifying the series accordingly.
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Percorso guidato
05:44Divergence Test (nth Term Test)
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Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.
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