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 / (n³ 3ⁿ) ]

Verified step by step guidance

1. Identify the given power series: \(\sum_{n=1}^{\infty} \frac{(x - 1)^n}{n^3 3^n}\).
2. To find the radius of convergence, apply the Root Test or Ratio Test. Here, the Ratio Test is convenient. Consider the general term \(a_n = \frac{(x - 1)^n}{n^3 3^n}\) and compute the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).
3. Calculate the ratio: \(\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(x - 1)^{n+1}}{(n+1)^3 3^{n+1}} \cdot \frac{n^3 3^n}{(x - 1)^n} \right| = \left| \frac{x - 1}{3} \right| \cdot \frac{n^3}{(n+1)^3}\).
4. Evaluate the limit as \(n \to \infty\): \(L = \left| \frac{x - 1}{3} \right| \cdot \lim_{n \to \infty} \frac{n^3}{(n+1)^3} = \left| \frac{x - 1}{3} \right|\).
5. The Ratio Test states the series converges if \(L < 1\), so the radius of convergence \(R\) satisfies \(\left| x - 1 \right| < 3\). This gives the interval \((1 - 3, 1 + 3)\) or \((-2, 4)\). Next, check convergence at the endpoints \(x = -2\) and \(x = 4\) by substituting back into the series and analyzing absolute and conditional convergence.

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. Finding these involves applying tests like the Ratio or Root Test to determine where the series converges absolutely.

Absolute Convergence

A series converges absolutely if the series of absolute values converges. This means ∑|a_n| converges, ensuring the original series converges regardless of term signs. Absolute convergence implies stronger convergence and often simplifies analysis of power series.

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. Identifying conditional convergence often requires testing endpoints of the interval of convergence separately.