Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?
∑ (from n = 1 to ∞) [ (x − 1)ⁿ / √n ]

Verified step by step guidance

1. Identify the given power series: \(\sum_{n=1}^{\infty} \frac{(x - 1)^n}{\sqrt{n}}\) and recognize that it is centered at \(x = 1\).
2. To find the radius of convergence, apply the Root Test or Ratio Test. Here, the Root Test is convenient: consider \(\lim_{n \to \infty} \sqrt[n]{\left| \frac{(x - 1)^n}{\sqrt{n}} \right|} = \lim_{n \to \infty} \frac{|x - 1|}{n^{1/(2n)}}\).
3. Since \(\lim_{n \to \infty} n^{1/(2n)} = 1\), the limit simplifies to \(|x - 1|\). For convergence, this limit must be less than 1, so the radius of convergence is \(R = 1\) and the interval of convergence is initially \(1 - 1 < x < 1 + 1\), or \((0, 2)\).
4. Next, check convergence at the endpoints \(x = 0\) and \(x = 2\) by substituting into the series: at \(x=0\), the series becomes \(\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}\); at \(x=2\), it becomes \(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}\). Analyze these using the Alternating Series Test and p-series test respectively.
5. Determine absolute convergence by testing \(\sum_{n=1}^{\infty} \left| \frac{(x - 1)^n}{\sqrt{n}} \right| = \sum_{n=1}^{\infty} \frac{|x - 1|^n}{\sqrt{n}}\). This series converges absolutely when \(|x - 1| < 1\). For conditional convergence, check if the series converges at endpoints where absolute convergence fails.

Key Concepts

Radius and Interval of Convergence

The radius of convergence is the distance from the center of a power series within which the series converges. The interval of convergence includes all x-values for which the series converges, possibly including endpoints. It is found using tests like the Ratio or Root Test applied to the general term.

Absolute Convergence

A series converges absolutely if the series of absolute values converges. This means ∑|a_n| converges, which guarantees convergence regardless of term signs. Absolute convergence implies regular convergence and is often easier to test using comparison or root tests.

Conditional Convergence

Conditional convergence occurs when a series converges, but does not converge absolutely. This means the series ∑a_n converges, but ∑|a_n| diverges. It often happens at the endpoints of the interval of convergence and requires tests like the Alternating Series Test.