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


∑ₖ₌₀∞ (x/3)ᵏ

Verified step by step guidance

1. Identify the given power series: \( \sum_{k=0}^{\infty} \left( \frac{x}{3} \right)^k \). This is a geometric series with the common ratio \( r = \frac{x}{3} \).
2. Recall that a geometric series \( \sum_{k=0}^{\infty} r^k \) converges if and only if \( |r| < 1 \). Apply this condition to the given series: \( \left| \frac{x}{3} \right| < 1 \).
3. Solve the inequality \( \left| \frac{x}{3} \right| < 1 \) to find the interval of convergence. Multiply both sides by 3 to get \( |x| < 3 \). This means the radius of convergence \( R = 3 \).
4. Express the interval of convergence as \( (-3, 3) \). Next, check the endpoints \( x = -3 \) and \( x = 3 \) by substituting them back into the series to determine if the series converges at these points.
5. At \( x = 3 \), the series becomes \( \sum_{k=0}^{\infty} 1^k \), which diverges. At \( x = -3 \), the series becomes \( \sum_{k=0}^{\infty} (-1)^k \), which also diverges. Therefore, the interval of convergence is \( (-3, 3) \) without including the endpoints.

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 power series involves recognizing how the variable x affects convergence depending on its distance from the center.

Radius of Convergence

The radius of convergence is the distance from the center within which a power series converges absolutely. It can be found using tests like the Ratio Test, and it defines the interval where the series behaves well.

Interval of Convergence

The interval of convergence is the set of x-values for which the power series converges. It includes all points within the radius of convergence and requires checking endpoints separately to determine if they are included.