Skip to main content
Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 11 - Power SeriesProblema 11.4.4
Capitolo 11, Problema 11.4.4

Use the Taylor series for cos x centered at 0 to verify that lim ₓ→ₐ (1− cos x)/x = 0.

Guida verificata passo dopo passo
1
Recall the Taylor series expansion of \( \cos x \) centered at 0, which is given by: \[ \cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \cdots \]
Substitute the Taylor series expansion of \( \cos x \) into the expression \( \frac{1 - \cos x}{x} \): \[ \frac{1 - \cos x}{x} = \frac{1 - \left(1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \cdots \right)}{x} \]
Simplify the numerator by distributing the negative sign and combining like terms: \[ 1 - 1 + \frac{x^{2}}{2!} - \frac{x^{4}}{4!} + \cdots = \frac{x^{2}}{2!} - \frac{x^{4}}{4!} + \cdots \]
Rewrite the expression as: \[ \frac{\frac{x^{2}}{2!} - \frac{x^{4}}{4!} + \cdots}{x} = \frac{x^{2}}{2! \cdot x} - \frac{x^{4}}{4! \cdot x} + \cdots = \frac{x}{2!} - \frac{x^{3}}{4!} + \cdots \]
Now, analyze the limit as \( x \to 0 \): each term contains a factor of \( x \) or higher powers of \( x \), so all terms approach 0. Therefore, \[ \lim_{x \to 0} \frac{1 - \cos x}{x} = 0 \]

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Taylor Series Expansion

The Taylor series expresses a function as an infinite sum of terms calculated from its derivatives at a single point. For cos x centered at 0, it expands as 1 - x²/2! + x⁴/4! - ..., which approximates cos x near zero and helps analyze limits involving cos x.
Video consigliato:
08:42
Taylor Series

Limit of a Function as x Approaches a Point

A limit describes the value a function approaches as the input approaches a specific point. Evaluating limₓ→0 (1 - cos x)/x involves understanding how the numerator and denominator behave near zero to determine the limit's value.
Video consigliato:
06:11
Limits of Rational Functions: Denominator = 0

Behavior of Higher-Order Terms in Limits

When using series expansions in limits, higher-order terms (like x⁴ and beyond) become negligible as x approaches zero. Recognizing which terms dominate allows simplification of expressions to find the limit accurately.
Video consigliato:
02:42
Higher Order Derivatives
Pratica correlata
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.


Find the Taylor polynomials p₁, …, p₅ centered at a=0 for f(x)=e⁻ˣ

37
views
Domanda del libro di testo

{Use of Tech} Approximations with Taylor polynomials


a. Approximate the given quantities using Taylor polynomials with n = 3.


b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.


e⁰ᐧ¹²

67
views
Domanda del libro di testo

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.


∑ₖ₌₀∞ (x/3)ᵏ

42
views
Domanda del libro di testo

{Use of Tech} Approximations with Taylor polynomials


a. Approximate the given quantities using Taylor polynomials with n = 3.


b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.


√1.06

83
views
Domanda del libro di testo

Evaluating an infinite series Let f(x) = (eˣ − 1)/x, for x ≠ 0, and f(0)=1. Use the Taylor series for f centered at 0 to evaluate f(1) and to find the value of ∑ₖ₌₀∞ 1/(k+1)!

54
views
Domanda del libro di testo

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.


∑ₖ₌₂∞ ((x+3)ᵏ)/(k łn²k)

46
views