Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (k²⁰ xᵏ)/(2k+1)!

Verified step by step guidance

1. Identify the given power series: \(\sum_{k=0}^{\infty} \frac{k^{20} x^{k}}{(2k+1)!}\). We want to find its radius and interval of convergence with respect to \(x\).
2. Use the Ratio Test to determine the radius of convergence. Consider the ratio of the absolute values of consecutive terms: \(\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\), where \(a_k = \frac{k^{20} x^{k}}{(2k+1)!}\).
3. Write the ratio explicitly: \(\left| \frac{(k+1)^{20} x^{k+1} / (2(k+1)+1)!}{k^{20} x^{k} / (2k+1)!} \right| = |x| \cdot \frac{(k+1)^{20}}{k^{20}} \cdot \frac{(2k+1)!}{(2k+3)!}\).
4. Simplify the factorial expression: \(\frac{(2k+1)!}{(2k+3)!} = \frac{1}{(2k+2)(2k+3)}\). Then analyze the limit as \(k \to \infty\) of the entire expression to find the radius of convergence \(R\).
5. Once the radius \(R\) is found, determine the interval of convergence by testing the endpoints \(x = -R\) and \(x = R\) separately to check if the series converges or diverges at those points.

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 or Root Test, which analyze the behavior of the series' terms as k approaches infinity.

Interval of Convergence

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