Textbook Question
Applying the Integral Test
Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.
∑ (from n = 2 to ∞) ln(n²) / n
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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- 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:
\[ \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \cdots \]
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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.
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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.
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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.
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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 = 2,aₙ₊₁ = 72 / (1 + aₙ)
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Textbook Question
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = nπ cos(nπ)
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Textbook Question
Uniqueness of limits Prove that limits of sequences are unique. That is, show that if L₁ and L₂ are numbers such that aₙ → L₁ and aₙ → L₂, then L₁ = L₂.
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Textbook Question
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (n + 3) / (n² + 5n + 6)
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Textbook Question
Find the value of b for which
1 + eᵇ + e²ᵇ + e³ᵇ + ⋯ = 9.