Radius and interval of convergence Determine the radius and interval of convergence of the following power series.


∑ₖ₌₁∞ (3x + 2)ᵏ/k

Verified step by step guidance

1. Identify the given power series: \( \sum_{k=1}^{\infty} \frac{(3x + 2)^k}{k} \). This is a power series centered at the point where the expression inside the power is zero, so first find the center by solving \(3x + 2 = 0\).
2. Solve for \(x\) to find the center of the series: \(3x + 2 = 0 \implies x = -\frac{2}{3}\). This means the series is centered at \(x = -\frac{2}{3}\).
3. To find the radius of convergence, use the root test or ratio test. Here, the root test is convenient. Consider the general term \(a_k = \frac{(3x + 2)^k}{k}\). The root test involves calculating \( \lim_{k \to \infty} \sqrt[k]{|a_k|} \).
4. Calculate the limit: \( \lim_{k \to \infty} \sqrt[k]{\left| \frac{(3x + 2)^k}{k} \right|} = \lim_{k \to \infty} \frac{|3x + 2|}{\sqrt[k]{k}} = |3x + 2| \), since \( \sqrt[k]{k} \to 1 \) as \(k \to \infty\).
5. Set the limit less than 1 for convergence: \( |3x + 2| < 1 \). Solve this inequality for \(x\) to find the interval of convergence. Then, check the endpoints \( |3x + 2| = 1 \) separately by substituting back into the original series to determine if the series converges or diverges at those points.

Key Concepts

Power Series and Convergence

A power series is an infinite sum of terms in the form a_k(x - c)^k, where c is the center. Understanding convergence means determining for which values of x the series sums to a finite value. This is essential to analyze the behavior of the series and find where it converges.

Radius of Convergence

The radius of convergence is the distance from the center c within which the power series converges absolutely. It can be found using tests like the Ratio Test or Root Test. Knowing the radius helps identify the interval on the x-axis 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 for convergence. This interval defines the domain where the series represents a valid function.