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 11.3.67e
Capitolo 11, Problema 11.3.67e

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
e. The Taylor series for an even function centered at 0 has only even powers of x.

Guida verificata passo dopo passo
1
Recall the definition of an even function: a function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in its domain.
Consider the Taylor series of \( f(x) \) centered at 0, which is given by \( f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \).
Substitute \( -x \) into the Taylor series: \( f(-x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} (-x)^n = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} (-1)^n x^n \).
Since \( f \) is even, \( f(-x) = f(x) \), so the series must satisfy \( \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} (-1)^n x^n = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \). This implies that terms with odd powers \( n \) must have zero coefficients because \( (-1)^n = -1 \) for odd \( n \), which would otherwise change the sign.
Therefore, the Taylor series for an even function centered at 0 contains only even powers of \( x \), confirming the statement is true.

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.

Even Functions

An even function satisfies f(-x) = f(x) for all x in its domain. This symmetry about the y-axis means the function's graph is mirrored on both sides, influencing the behavior of its Taylor series expansion.
Video consigliato:
6:13
Exponential Functions

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. Centering at zero (Maclaurin series) expresses the function as powers of x, revealing patterns in coefficients based on the function's properties.
Video consigliato:
08:42
Taylor Series

Parity of Terms in Taylor Series for Even Functions

For even functions centered at zero, all odd-powered terms in the Taylor series vanish because the derivatives of odd order at zero are zero. This results in a series containing only even powers of x, reflecting the function's symmetry.
Video consigliato:
08:42
Taylor Series
Pratica correlata
Domanda del libro di testo

{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals

S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt

e. How many terms of the Maclaurin series are required to approximate C(−0.25) with an error no greater than 10⁻⁶?

68
views
Domanda del libro di testo

{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals

S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt

d. How many terms of the Maclaurin series are required to approximate S(0.05) with an error no greater than 10⁻⁴?

95
views
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. If p(x) is the Taylor series for f centered at 0, then p(x−1) is the Taylor series for f centered at 1.

43
views
Domanda del libro di testo

Matching functions with polynomials Match functions a–f with Taylor polynomials A–F (all centered at 0). Give reasons for your choices.


f. e⁻²ˣ


A. p₂(x)= 1 + 2x + 2x²

B. p₂(x) = 1 − 6x + 24x²

C. p₂(x) = 1 + x − x²/2

D. p₂(x) = 1 − 2x + 4x²

E. p₂(x) = 1 − x + (3/2)x²

F. p₂(x) = 1 − 2x + 2x²

41
views
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


If f(x)=∑ₖ₌₀∞ cₖ xᵏ=0, for all x on an interval (−a, a), then cₖ = 0, for all k.

62
views