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.R.65d
Capitolo 11, Problema 11.R.65d

Approximating ln 2 Consider the following three ways to approximate
ln 2.
d. At what value of x should the series in part (c) be evaluated to approximate ln 2? Write the resulting infinite series for ln 2.

Guida verificata passo dopo passo
1
Recall that the natural logarithm function \( \ln(1+x) \) can be expressed as a power series centered at \( x=0 \), given by the Taylor series: \[ \ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \]
To approximate \( \ln 2 \), recognize that \( \ln 2 = \ln(1 + 1) \). This means we want to evaluate the series at \( x = 1 \).
Substitute \( x = 1 \) into the series to write the infinite series representation for \( \ln 2 \): \[ \ln 2 = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{1^n}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots \]
This alternating series converges to \( \ln 2 \), and each partial sum provides an approximation of \( \ln 2 \).
Thus, the value of \( x \) to evaluate the series at is \( x = 1 \), and the resulting infinite series for \( \ln 2 \) is the alternating harmonic series as shown above.

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:
6m

Concetti chiave

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

Taylor and Maclaurin Series

Taylor and Maclaurin series express functions as infinite sums of polynomial terms based on derivatives at a specific point. The Maclaurin series is a special case centered at zero. These series allow approximation of functions like ln(x) near a point by summing terms involving powers of (x - a).
Video consigliato:
08:26
Convergence of Taylor & Maclaurin Series

Choosing the Expansion Point

Selecting the value of x (or the center a) for evaluating a series is crucial for accurate approximation. For ln 2, choosing x close to 1 or another convenient point simplifies the series and improves convergence. The choice affects the form and convergence speed of the infinite series.
Video consigliato:
Percorso guidato
07:51
Choosing a Convergence Test

Infinite Series Representation of ln(2)

The natural logarithm ln(2) can be represented as an infinite series derived from the Taylor or Maclaurin expansion of ln(1 + x). For example, ln(2) = ln(1 + 1) can be expressed as the alternating series 1 - 1/2 + 1/3 - 1/4 + ..., which converges to ln(2). Understanding this series helps in approximating ln(2) numerically.
Video consigliato:
Percorso guidato
06:52
Convergence of an Infinite Series
Pratica correlata
Domanda del libro di testo

Limits by power series Use Taylor series to evaluate the following limits.


lim ₙ → 0 (x²/2 - 1 + cos x)/x⁴

89
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. There is more than one way to choose the center of the series.


sinh (-1)

72
views
Domanda del libro di testo

Approximating ln 2 Consider the following three ways to approximate

ln 2.

c. Use the property ln a/b = ln a - ln b and the series of parts (a) and (b) to find the Taylor series for ƒ(x) = ln (1 + x)/(1 - x) b centered at 0.

38
views
Domanda del libro di testo

Power series from the geometric series Use the geometric series a Σₖ ₌ ₀ ∞ (x)ᵏ = 1/(1 - x), for |x| < 1, to determine the Maclaurin series and the interval of convergence for the following functions.


ƒ(x) = 1/(1 + 5x)

101
views
Domanda del libro di testo

A differential equation Find a power series solution of the differential equation y'(x) - 4y + 12 = 0, subject to the condition y(0) = 4. Identify the solution in terms of known functions.

124
views
Domanda del libro di testo

Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.

ƒ(x) = cos x, a = π/2

86
views