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

Verified step by step guidance

1. Identify the given power series: \( \sum_{k=1}^{\infty} \frac{(-1)^{k+1} (x-1)^k}{k} \). This is a power series centered at \( x = 1 \).
2. To find the radius of convergence, use the Ratio Test or Root Test. Here, the Ratio Test is convenient. Consider the general term \( a_k = \frac{(-1)^{k+1} (x-1)^k}{k} \).
3. Apply the Ratio Test by computing \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \left| \frac{(x-1)^{k+1}}{k+1} \cdot \frac{k}{(x-1)^k} \right| = \lim_{k \to \infty} |x-1| \cdot \frac{k}{k+1} = |x-1| \).
4. 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 \( x \in (0, 2) \).
5. 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.

Key Concepts

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.

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.

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.