Skip to main content
Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblema 10.9.16
Capitolo 10, Problema 10.9.16

Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.
sin x – x + (x³ / 3!)

Guida verificata passo dopo passo
1
Recall the Taylor series expansion of \( \sin x \) at \( x = 0 \), which is given by: \[ \sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \]
Write down the given expression explicitly: \[ \sin x - x + \frac{x^3}{3!} \]
Substitute the Taylor series for \( \sin x \) into the expression: \[ \left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \right) - x + \frac{x^3}{3!} \]
Combine like terms by grouping powers of \( x \): Notice that \( x \) and \( -x \) cancel out, and \( -\frac{x^3}{3!} \) and \( +\frac{x^3}{3!} \) also cancel out, leaving the series starting from the \( x^5 \) term:
Write the simplified Taylor series starting from the \( x^5 \) term: \[ \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \cdots \]

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:
2m

Concetti chiave

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

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, often x = 0 (Maclaurin series). It approximates functions locally and is essential for expressing functions like sin x as power series.
Video consigliato:
08:42
Taylor Series

Power Series Operations

Power series operations include addition, subtraction, multiplication, and differentiation of series. These operations allow manipulation of known series to find new series representations, such as combining or modifying the Taylor series of sin x to match the given expression.
Video consigliato:
05:58
Intro to Power Series

Maclaurin Series for sin x

The Maclaurin series for sin x is an alternating series of odd powers of x with factorial denominators: sin x = x - x³/3! + x⁵/5! - ... . Understanding this series is crucial to simplifying or adjusting terms like sin x – x + (x³/3!) in the problem.
Video consigliato:
08:26
Convergence of Taylor & Maclaurin Series
Pratica correlata
Domanda del libro di testo

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) sin (1/n)

29
views
Domanda del libro di testo

Recursively Defined Sequences

In Exercises 101–108, assume that each sequence converges and find its limit.

a₁ = 5,aₙ₊₁ = √(5aₙ)

27
views
Domanda del libro di testo

Finding nth Partial Sums

In Exercises 1–6, find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges.

2 + (2/3) + (2/9) + (2/27) + … + (2 / 3ⁿ⁻¹) + …

4
views
Domanda del libro di testo

Determining Convergence or Divergence

In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑(from n=1 to ∞) [n! / (2n + 1)!]

Domanda del libro di testo

In Exercises 35–40, find the first three nonzero terms of the Maclaurin series for each function.

f(x) = (sin x) ln(1 + x)

Domanda del libro di testo

In Exercises 81–86, find the values of x for which the given geometric series converges.

Also, find the sum of the series (as a function of x) for those values of x.

∑ (from n = 0 to ∞) [ (−1/2)ⁿ (x − 3)ⁿ ]