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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.2.32
Chapter 11, Problem 11.2.32

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (-x/10)²ᵏ

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1
Rewrite the given power series in a more standard form. The series is \( \sum_{k=0}^\infty \left(-\frac{x}{10}\right)^{2k} \). Notice that \( \left(-\frac{x}{10}\right)^{2k} = \left(\frac{x^2}{100}\right)^k \). So the series can be expressed as \( \sum_{k=0}^\infty \left(\frac{x^2}{100}\right)^k \).
Recognize that this is a geometric series with common ratio \( r = \frac{x^2}{100} \). A geometric series \( \sum_{k=0}^\infty r^k \) converges if and only if \( |r| < 1 \).
Set up the inequality for convergence: \( \left| \frac{x^2}{100} \right| < 1 \). Since \( x^2 \geq 0 \), this simplifies to \( \frac{x^2}{100} < 1 \).
Solve the inequality for \( x \): multiply both sides by 100 to get \( x^2 < 100 \), then take the square root to find \( |x| < 10 \). This gives the radius of convergence \( R = 10 \).
Determine the interval of convergence by considering the endpoints \( x = -10 \) and \( x = 10 \). Substitute these values back into the original series and test for convergence at each endpoint to finalize the interval.

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Key Concepts

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

Power Series

A power series is an infinite sum of terms in the form a_k(x - c)^k, where a_k are coefficients and c is the center. Understanding the structure of power series is essential to analyze their convergence behavior depending on the variable x.
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Intro to Power Series

Radius of Convergence

The radius of convergence is the distance from the center within which a power series converges absolutely. It can be found using tests like the Ratio or Root Test, and it defines the interval where the series behaves well.
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Radius of Convergence

Interval of Convergence

The interval of convergence is the set of x-values for which the power series converges. It includes all points within the radius of convergence and requires checking endpoints separately to determine if they are included.
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Interval of Convergence
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