Textbook Question
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)
1
views
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 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!)
Verified step by step guidance1
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:
\[ \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \cdots \]
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
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.
Recommended video:
05:58Intro 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.
Recommended video:
08:26Convergence of Taylor & Maclaurin Series
Related Practice
Textbook Question
Recursively Defined Sequences
In Exercises 101–108, assume that each sequence converges and find its limit.
a₁ = 5,aₙ₊₁ = √(5aₙ)
Textbook Question
In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ (n!)² / (2ⁿ (2n)!) ] xⁿ
Textbook Question
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)!]
Textbook Question
In Exercises 35–40, find the first three nonzero terms of the Maclaurin series for each function.
f(x) = (sin x) ln(1 + x)
Textbook Question
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 ]