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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 11.2.30
Chapter 11, Problem 11.2.30

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ k(x−1)ᵏ

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Identify the given power series: \(\sum_{k=0}^{\infty} k (x - 1)^k\). This is a power series centered at \(x = 1\).
To find the radius of convergence, use the Ratio Test. Consider the general term \(a_k = k (x - 1)^k\). Compute the limit \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\).
Calculate the ratio inside the limit: \(\left| \frac{(k+1)(x-1)^{k+1}}{k (x-1)^k} \right| = \left| \frac{k+1}{k} \right| \cdot |x-1| = \left(1 + \frac{1}{k}\right) |x-1|\).
Take the limit as \(k \to \infty\): \(L = \lim_{k \to \infty} \left(1 + \frac{1}{k}\right) |x-1| = |x-1|\). The Ratio Test says the series converges if \(L < 1\), so the radius of convergence \(R\) satisfies \(|x-1| < 1\).
Determine the interval of convergence by checking the endpoints \(x = 1 - 1 = 0\) and \(x = 1 + 1 = 2\). Substitute these values into the series and test for convergence using appropriate tests (e.g., p-series, alternating series, or divergence test).

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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 around the center point.
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Intro to Power Series

Radius of Convergence

The radius of convergence is the distance from the center within which the power series converges absolutely. It can be found using tests like the Ratio Test 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 includes all x-values for which the power series converges, typically centered at c and extending radius R in both directions. Endpoints must be checked separately to determine if the series converges or diverges there.
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Interval of Convergence
Related Practice
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