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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 10.3.2
Chapter 10, Problem 10.3.2

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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1
Identify the series given: \( \sum_{n=1}^{\infty} \frac{1}{n^{0.2}} \). We want to determine if this series converges or diverges using the Integral Test.
Recall the Integral Test conditions: The function \( f(x) = \frac{1}{x^{0.2}} \) must be positive, continuous, and decreasing for \( x \geq 1 \). Verify these conditions for \( f(x) \).
Set up the improper integral corresponding to the series: \( \int_1^{\infty} \frac{1}{x^{0.2}} \, dx \). This integral will help determine the behavior of the series.
Evaluate the integral \( \int_1^{\infty} x^{-0.2} \, dx \) by finding the antiderivative of \( x^{-0.2} \), which involves increasing the exponent by 1 and dividing by the new exponent.
Analyze the limit of the integral as the upper bound approaches infinity. If the integral converges (has a finite value), then the series converges; if it diverges (goes to infinity), then the series diverges.

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Key Concepts

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

Integral Test for Convergence

The Integral Test determines the convergence or divergence of an infinite series by comparing it to an improper integral. If the function f(x) corresponding to the series terms is positive, continuous, and decreasing for x ≥ 1, then the series ∑ f(n) and the integral ∫ f(x) dx from 1 to ∞ either both converge or both diverge.
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Conditions for Applying the Integral Test

To apply the Integral Test, the function f(x) must be positive, continuous, and decreasing on the interval [1, ∞). Verifying these conditions ensures the test is valid. For the series ∑ 1/n^0.2, checking these properties for f(x) = 1/x^0.2 is essential before proceeding.
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Integral Test

Behavior of p-Series and Improper Integrals

The series ∑ 1/n^p is known as a p-series, which converges if p > 1 and diverges otherwise. Similarly, the integral ∫ 1/x^p dx from 1 to ∞ converges if p > 1. Understanding this helps quickly determine the convergence of the given series with p = 0.2, which is less than 1, indicating divergence.
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Improper Integrals: Infinite Intervals
Related Practice
Textbook Question

In Exercises 121–124, determine whether the sequence is monotonic and whether it is bounded.

aₙ = 2ⁿ 3ⁿ / n!

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Textbook Question

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

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 ∞) (1 − n) / n2ⁿ

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Textbook Question

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

∑ (from n = 1 to ∞) [ n! xⁿ / nⁿ ]

Textbook Question

Use series to evaluate the limits in Exercises 29–40.

37. lim (x → 0) ln(1 + x²) / (1 - cos(x))

Textbook Question

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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