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

∑ₖ₌₂∞ ((x+3)ᵏ)/(k łn²k)

검증된 단계별 안내

Identify the given power series: \( \sum_{k=2}^{\infty} \frac{(x+3)^k}{k \ln^2(k)} \). Notice that the series is centered at \( x = -3 \) because the term is \( (x+3)^k \).

To find the radius of convergence, apply the Root Test or the Ratio Test. Here, the Root Test is convenient. Consider the general term \( a_k = \frac{(x+3)^k}{k \ln^2(k)} \). Compute \( \lim_{k \to \infty} \sqrt[k]{|a_k|} \).

Calculate \( \sqrt[k]{|a_k|} = \sqrt[k]{\frac{|x+3|^k}{k \ln^2(k)}} = |x+3| \cdot \sqrt[k]{\frac{1}{k \ln^2(k)}} \). As \( k \to \infty \), \( \sqrt[k]{k} \to 1 \) and \( \sqrt[k]{\ln^2(k)} \to 1 \), so the limit simplifies to \( |x+3| \).

The Root Test states that the series converges if \( \lim_{k \to \infty} \sqrt[k]{|a_k|} < 1 \), so the radius of convergence \( R \) satisfies \( |x+3| < 1 \). Thus, the radius of convergence is \( R = 1 \).

Next, determine the interval of convergence by checking the endpoints \( x = -3 - 1 = -4 \) and \( x = -3 + 1 = -2 \). Substitute these values into the original series and analyze convergence using appropriate tests (such as the Comparison Test or Integral Test) because the behavior at endpoints depends on the series without the \( (x+3)^k \) factor.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

주요 개념

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

Radius of Convergence

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

Interval of Convergence

The interval of convergence is the set of all x-values for which the power series converges. It includes the radius of convergence and requires checking endpoints separately, as convergence at these points is not guaranteed by the radius alone.

Ratio Test for Convergence

The Ratio Test determines convergence by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges; if greater than one, it diverges. It is especially useful for power series to find the radius of convergence.

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.∑ₖ₌₂∞ ((x+3)ᵏ)/(k łn²k)