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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblema 10.10.37
Capitolo 10, Problema 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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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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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
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Finding Taylor and Maclaurin Series

In Exercises 25–34, find the Taylor series generated by f at x = a.

f(x) = cos(2x + π/2),a = π/4

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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 ∞) (1 − n) / n2ⁿ

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Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 1 to ∞) [ (x − 1)ⁿ / (n³ 3ⁿ) ]

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

∑ (from n = 1 to ∞) [ n! xⁿ / nⁿ ]

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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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Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 0 to ∞) [ n xⁿ / (4ⁿ (n² + 1)) ]

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