Domanda del libro di testo
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.
f(x) = 2x/(1 + x²)²
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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 11.4.8
Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (tan ⁻¹ x − x)/x³"
Guida verificata passo dopo passo1
Recognize that the problem asks to evaluate the limit \( \lim_{x \to 0} \frac{\tan^{-1} x - x}{x^3} \) using Taylor series expansions.
Recall the Taylor series expansion of \( \tan^{-1} x \) around \( x = 0 \):
\[
\tan^{-1} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots
\]
Substitute the Taylor series expansion into the numerator:
\[
\tan^{-1} x - x = \left(x - \frac{x^3}{3} + \cdots \right) - x = - \frac{x^3}{3} + \cdots
\]
Rewrite the original limit expression using this substitution:
\[
\lim_{x \to 0} \frac{\tan^{-1} x - x}{x^3} = \lim_{x \to 0} \frac{- \frac{x^3}{3} + \cdots}{x^3}
\]
Simplify the fraction by dividing each term by \( x^3 \), then evaluate the limit by letting \( x \to 0 \), which will eliminate higher order terms, leaving the coefficient of the leading term.
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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 to be simplified into polynomials. For limits, Taylor expansions help identify dominant terms and simplify evaluation.
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08:42
Taylor Series
Inverse Tangent Function (arctan) Properties
The inverse tangent function, arctan(x), is smooth and differentiable around zero, with a known Taylor series expansion. Understanding its series helps express arctan(x) as x minus higher-order terms, which is essential for evaluating limits involving arctan(x) near zero.
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3:17Inverse Tangent
Limit Evaluation Using Series Expansion
When direct substitution in a limit leads to an indeterminate form, expanding functions into their Taylor series can reveal the behavior of the numerator and denominator. By comparing the lowest-order nonzero terms, one can compute the limit accurately without complex algebraic manipulation.
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Percorso guidato
06:45Intro to Series: Partial Sums
Pratica correlata
Domanda del libro di testo
Series to functions Find the function represented by the following series, and find the interval of convergence of the series. (Not all these series are power series.)
∑ₖ₌₀∞ e⁻ᵏˣ
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Domanda del libro di testo
Series to functions Find the function represented by the following series, and find the interval of convergence of the series. (Not all these series are power series.)
∑ₖ₌₀∞(√x − 2)ᵏ
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Domanda del libro di testo
Use of Tech Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at a.
c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
f(x)=e⁻²ˣ, a=0; approximate e⁻⁰ᐧ².
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Domanda del libro di testo
Exponential function In Section 11.3, we show that the power series for the exponential function centered at 0 is
eˣ = ∑ₖ₌₀∞ (xᵏ)/k!, for −∞ < x < ∞
Use the methods of this section to find the power series centered at 0 for the following functions. Give the interval of convergence for the resulting series.
f(x) = x²eˣ
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Domanda del libro di testo
Radius of convergence Find the radius of convergence for the following power series.
∑ₖ₌₁∞ (1−cos (1/2ᵏ)) xᵏ
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