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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.10.37
Chapter 10, Problem 10.10.37

Use series to evaluate the limits in Exercises 29–40.
37. lim (x → 0) ln(1 + x²) / (1 - cos(x))

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1
Recall the Maclaurin series expansions for the functions involved: \( \ln(1 + x^2) \) and \( \cos(x) \).
Write the series expansion for \( \ln(1 + x^2) \) around \( x = 0 \). Since \( \ln(1 + u) = u - \frac{u^2}{2} + \frac{u^3}{3} - \cdots \), substitute \( u = x^2 \) to get \( \ln(1 + x^2) = x^2 - \frac{x^4}{2} + \frac{x^6}{3} - \cdots \).
Write the series expansion for \( \cos(x) \) around \( x = 0 \): \( \cos(x) = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \).
Substitute the series expansions into the original limit expression: \( \frac{\ln(1 + x^2)}{1 - \cos(x)} = \frac{x^2 - \frac{x^4}{2} + \cdots}{1 - \left(1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \right)} \).
Simplify the denominator: \( 1 - \cos(x) = \frac{x^2}{2} - \frac{x^4}{24} + \cdots \). Then, divide the numerator series by the denominator series, and identify the leading terms to find the limit as \( x \to 0 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Taylor and Maclaurin Series

Taylor and Maclaurin series express functions as infinite sums of powers of (x - a) or x, respectively. They allow approximation of functions near a point, making it easier to analyze limits, derivatives, and integrals. For example, ln(1 + x²) and cos(x) can be expanded into power series around x = 0.
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Convergence of Taylor & Maclaurin Series

Limit Evaluation Using Series Expansion

When direct substitution in a limit leads to an indeterminate form, series expansions help by approximating numerator and denominator with polynomials. This simplification often reveals the limit's value by canceling terms or comparing leading powers, especially useful for limits involving logarithmic and trigonometric functions.
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Intro to Series: Partial Sums

Behavior of Logarithmic and Trigonometric Functions Near Zero

Understanding how ln(1 + x²) and cos(x) behave near zero is crucial. ln(1 + x²) behaves like x² for small x, while cos(x) approximates 1 - x²/2. Recognizing these approximations helps simplify the limit expression and identify dominant terms in numerator and denominator.
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

Determining Convergence or Divergence

Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

∑ (from n=1 to ∞) eⁿ / (1 + e²ⁿ)

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Textbook Question

Finding Terms of a Sequence

Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.

aₙ = 1 / n!

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Textbook Question

Power Series

In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.

∑ (from n = 1 to ∞) (csch n)xⁿ

Textbook Question

In Exercises 37–42, find the series’ radius of convergence.

∑ (from n = 1 to ∞) [ n! xⁿ / nⁿ ]

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 = 1 to ∞) 1 / n⁰·²

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Textbook Question

30. b. By differentiating the series in part (a) term by term, show that

Σ(from n=1 to ∞) n / (n + 1)! = 1.