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

∑ₖ₌₁∞ (xᵏ/kᵏ)

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Identify the given power series: \(\displaystyle \sum_{k=1}^{\infty} \frac{x^{k}}{k^{k}}\).

To find the radius of convergence, apply the Root Test, which involves computing \(\lim_{k \to \infty} \sqrt[k]{\left| \frac{x^{k}}{k^{k}} \right|} = \lim_{k \to \infty} \frac{|x|}{k}\).

Evaluate the limit: since \(\lim_{k \to \infty} \frac{|x|}{k} = 0\) for all real \(x\), the Root Test tells us the series converges for all \(x\).

Conclude that the radius of convergence \(R\) is infinite, meaning \(R = \infty\).

Since the radius of convergence is infinite, the interval of convergence is \((-\infty, \infty)\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 root or 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 separate checking of endpoints to determine if they are included.

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.∑ₖ₌₁∞ (xᵏ/kᵏ)