Domanda del libro di testo
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
61
views
Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 11.2.11
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₁∞ (kx)ᵏ
Guida verificata passo dopo passo1
Identify the given power series: \( \sum_{k=1}^{\infty} (kx)^k \). This can be rewritten as \( \sum_{k=1}^{\infty} k^k x^k \).
To find the radius of convergence, use the root test which involves the limit \( L = \lim_{k \to \infty} \sqrt[k]{|a_k|} \), where \( a_k = k^k x^k \).
Calculate \( \sqrt[k]{|a_k|} = \sqrt[k]{k^k |x|^k} = k |x| \).
Find the limit \( L = \lim_{k \to \infty} k |x| \). For the series to converge, this limit must be less than 1, so set \( \lim_{k \to \infty} k |x| < 1 \).
Since \( \lim_{k \to \infty} k |x| = \infty \) for any \( x \neq 0 \), the radius of convergence is 0, meaning the series converges only at \( x = 0 \). The interval of convergence is therefore \( \{0\} \).
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:
4mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Power Series
A power series is an infinite sum of terms in the form a_k(x - c)^k, where a_k are coefficients and c is the center. Understanding the structure of power series is essential to analyze their convergence behavior depending on the variable x.
Video consigliato:
05:58
Intro to Power Series
Radius of Convergence
The radius of convergence is the distance from the center c within which the power series converges absolutely. It can be found using tests like the Ratio Test or Root Test, and it defines the interval where the series behaves well.
Video consigliato:
07:36Radius of Convergence
Interval of Convergence
The interval of convergence is the set of all x-values for which the power series converges. It includes all points within the radius of convergence and requires checking endpoints separately to determine if the series converges there.
Video consigliato:
08:44Interval of Convergence
Pratica correlata
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) = ln √(4 − x²)
60
views
Domanda del libro di testo
Shifting power series If the power series f(x)=∑ cₖ xᵏ has an interval of convergence of |x|<R, what is the interval of convergence of the power series for f(x−a), where a ≠ 0 is a real number?
44
views
Domanda del libro di testo
{Use of Tech} Best center point Suppose you wish to approximate cos (π/ 2) using Taylor polynomials. Is the approximation more accurate if you use Taylor polynomials centered at 0 or at π/6? Use a calculator for numerical experiments and check for consistency with Theorem 11.2. Does the answer depend on the order of the polynomial?
48
views
Domanda del libro di testo
Power series for derivatives
a. Differentiate the Taylor series centered at 0 for the following functions.
b. Identify the function represented by the differentiated series.
c. Give the interval of convergence of the power series for the derivative.
f(x) = (1 − x)⁻¹
61
views
Domanda del libro di testo
Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.
√e
68
views