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


∑ₖ₌₀∞ (2x)ᵏ/k!

Verified step by step guidance

1. Identify the given power series: \( \sum_{k=0}^{\infty} \frac{(2x)^k}{k!} \). This is a power series centered at 0 with general term \( a_k = \frac{(2x)^k}{k!} \).
2. To find the radius of convergence, use the Ratio Test. Consider the limit \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). Substitute \( a_k \) and \( a_{k+1} \): \[ L = \lim_{k \to \infty} \left| \frac{(2x)^{k+1} / (k+1)!}{(2x)^k / k!} \right| = \lim_{k \to \infty} \left| \frac{(2x)^{k+1}}{(k+1)!} \cdot \frac{k!}{(2x)^k} \right| = \lim_{k \to \infty} \left| \frac{2x}{k+1} \right| \]
3. Evaluate the limit \( L \). Since \( \lim_{k \to \infty} \frac{2|x|}{k+1} = 0 \), the Ratio Test tells us the series converges for all real \( x \).
4. Because the limit \( L = 0 < 1 \) for all \( x \), the radius of convergence \( R = \infty \). This means the power series converges for every real number \( x \).
5. Therefore, the interval of convergence is \( (-\infty, \infty) \). No endpoints need to be checked since the series converges everywhere.

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 helps analyze functions represented as infinite polynomials and is essential for determining convergence properties.

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 or Root Test, indicating the interval where the series represents a valid function.

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 the series converges or diverges there.