Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (-x/10)²ᵏ

Verified step by step guidance

1. Rewrite the given power series in a more standard form. The series is \( \sum_{k=0}^\infty \left(-\frac{x}{10}\right)^{2k} \). Notice that \( \left(-\frac{x}{10}\right)^{2k} = \left(\frac{x^2}{100}\right)^k \). So the series can be expressed as \( \sum_{k=0}^\infty \left(\frac{x^2}{100}\right)^k \).
2. Recognize that this is a geometric series with common ratio \( r = \frac{x^2}{100} \). A geometric series \( \sum_{k=0}^\infty r^k \) converges if and only if \( |r| < 1 \).
3. Set up the inequality for convergence: \( \left| \frac{x^2}{100} \right| < 1 \). Since \( x^2 \geq 0 \), this simplifies to \( \frac{x^2}{100} < 1 \).
4. Solve the inequality for \( x \): multiply both sides by 100 to get \( x^2 < 100 \), then take the square root to find \( |x| < 10 \). This gives the radius of convergence \( R = 10 \).
5. Determine the interval of convergence by considering the endpoints \( x = -10 \) and \( x = 10 \). Substitute these values back into the original series and test for convergence at each endpoint to finalize the interval.

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 the structure of power series is essential to analyze their convergence behavior depending on the variable x.

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 or Root 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.