Domanda del libro di testo
Evaluating an infinite series Write the Maclaurin series for f(x) = ln (1+x) and find the interval of convergence. Evaluate f(−1/2) to find the value of ∑ₖ₌₁∞ 1/(k 2ᵏ)
116
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 11.2.42
Combining power series Use the geometric series
f(x) = 1/(1-x) = ∑ₖ₌₀∞ xᵏ, for |x| < 1,
to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.
g(x) = x³/(1 − x)
Guida verificata passo dopo passo1
Recall the given geometric series formula: \[f(x) = \frac{1}{1 - x} = \sum_{k=0}^{\infty} x^{k}, \quad \text{for } |x| < 1.\] This series converges when the absolute value of \(x\) is less than 1.
To find the power series representation for \[g(x) = \frac{x^{3}}{1 - x},\] notice that it can be written as \[g(x) = x^{3} \cdot \frac{1}{1 - x}.\]
Substitute the power series for \[\frac{1}{1 - x}\] into the expression for \[g(x)\]: \[g(x) = x^{3} \sum_{k=0}^{\infty} x^{k}.\]
Use properties of exponents to combine the powers of \[x\] inside the summation: \[g(x) = \sum_{k=0}^{\infty} x^{k + 3} = \sum_{k=0}^{\infty} x^{k + 3}.\]
Rewrite the series with a new index if desired (for example, let \[n = k + 3\]) to express the series in standard form. The interval of convergence remains \[|x| < 1\] because multiplying by \[x^{3}\] does not affect convergence.
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:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Geometric Series and Its Power Series Representation
A geometric series is a sum of the form ∑ x^k for k from 0 to infinity, which converges to 1/(1 - x) when |x| < 1. This fundamental series allows us to express many functions as power series by manipulating the variable or coefficients. Understanding this series is key to rewriting functions like g(x) in terms of power series.
Video consigliato:
Percorso guidato
06:00
Geometric Series
Manipulating Power Series to Represent New Functions
To find the power series of a function like g(x) = x³/(1 - x), we use substitution and multiplication on the known geometric series. Multiplying the entire series by x³ shifts the powers, resulting in a new series ∑ x^(k+3). This technique helps generate power series for more complex functions based on simpler known series.
Video consigliato:
07:32Representing Functions as Power Series
Interval of Convergence for Power Series
The interval of convergence is the set of x-values for which a power series converges. For the geometric series 1/(1 - x), the interval is |x| < 1. When manipulating the series (e.g., multiplying by x³), the interval of convergence remains the same unless the function introduces new singularities. Identifying this interval ensures the series representation is valid.
Video consigliato:
08:44Interval of Convergence
Pratica correlata
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.
cos 2
67
views
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, p₂, and p₃ centered at a=1 for f(x)=x³.
55
views
Domanda del libro di testo
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₁∞ (x²ᵏ)/k
84
views
Domanda del libro di testo
Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (eˣ − e⁻ˣ)/x
87
views
Domanda del libro di testo
Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.
g(x) = 2/(1 − 2x)² using f(x) = 1/(1 − 2x)
84
views