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 ∞) [ (ln n) xⁿ ]

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Identify the given power series: \(\sum_{n=1}^{\infty} (\ln n) x^{n}\). We want to find its radius and interval of convergence.

Use the Root Test or Ratio Test to find the radius of convergence \(R\). For the Ratio Test, consider the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\) where \(a_n = (\ln n) x^n\).

Calculate the ratio: \(\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(\ln (n+1)) x^{n+1}}{(\ln n) x^n} \right| = \left| x \right| \cdot \frac{\ln (n+1)}{\ln n}\). Then find \(L = \lim_{n \to \infty} \left| x \right| \cdot \frac{\ln (n+1)}{\ln n}\).

Since \(\lim_{n \to \infty} \frac{\ln (n+1)}{\ln n} = 1\), the limit simplifies to \(L = |x|\). The Ratio Test says the series converges if \(L < 1\), so the radius of convergence is \(R = 1\).

Determine the interval of convergence by testing the endpoints \(x = -1\) and \(x = 1\) separately. For each endpoint, analyze the series \(\sum (\ln n) x^n\) to check for absolute and conditional convergence.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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. Absolute convergence guarantees convergence regardless of the sign of terms and implies stronger stability. For power series, checking absolute convergence often involves applying convergence tests to |a_n x^n|.

Conditional Convergence

Conditional convergence occurs when a series converges but does not converge absolutely. This means the series converges only because of the alternating nature or specific arrangement of terms. Identifying conditional convergence requires testing both the original and absolute value series.

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Related Practice

Textbook Question

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = (xⁿ / (2n + 1))^(1/n),x > 0

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Textbook Question

In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.

∑ (from n = 0 to ∞) [ (x² − 1) / 2 ]ⁿ