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

Verified step by step guidance

1. Identify the given power series: \(\sum_{k=0}^{\infty} k (x - 1)^k\). This is a power series centered at \(x = 1\).
2. To find the radius of convergence, use the Ratio Test. Consider the general term \(a_k = k (x - 1)^k\). Compute the limit \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\).
3. Calculate the ratio inside the limit: \(\left| \frac{(k+1)(x-1)^{k+1}}{k (x-1)^k} \right| = \left| \frac{k+1}{k} \right| \cdot |x-1| = \left(1 + \frac{1}{k}\right) |x-1|\).
4. 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 \(|x-1| < 1\).
5. 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).

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 the structure of power series is essential to analyze their convergence behavior around the center point.

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.

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.