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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 11 - Power SeriesProblema 11.2.23
Capitolo 11, Problema 11.2.23

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.


∑ₖ₌₁∞ ((−1)ᵏ⁺¹(x−1)ᵏ)/k

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Identify the given power series: \( \sum_{k=1}^{\infty} \frac{(-1)^{k+1} (x-1)^k}{k} \). This is a power series centered at \( x = 1 \).
To find the radius of convergence, use the Ratio Test or Root Test. Here, the Ratio Test is convenient. Consider the general term \( a_k = \frac{(-1)^{k+1} (x-1)^k}{k} \).
Apply the Ratio Test by computing \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \left| \frac{(x-1)^{k+1}}{k+1} \cdot \frac{k}{(x-1)^k} \right| = \lim_{k \to \infty} |x-1| \cdot \frac{k}{k+1} = |x-1| \).
The Ratio Test states the series converges if \( L < 1 \), so the radius of convergence \( R = 1 \). This means the series converges for \( |x-1| < 1 \), or \( x \in (0, 2) \).
Check the endpoints \( x=0 \) and \( x=2 \) by substituting into the series and testing for convergence (e.g., using the Alternating Series Test or p-series test) to determine the interval of convergence.

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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 power series helps analyze functions as infinite polynomials and is essential for studying 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 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 represents a valid function.
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Radius of Convergence

Interval of Convergence

The interval of convergence includes all x-values for which the power series converges, possibly including endpoints. After finding the radius, endpoint testing is necessary to determine if the series converges or diverges at those boundary points.
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Interval of Convergence
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