9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
∑ (from k = 1 to ∞) (2ᵏ / k⁹⁹)

검증된 단계별 안내

Identify the series given: \( \sum_{k=1}^{\infty} \frac{2^{k}}{k^{99}} \). We want to determine if it converges absolutely or diverges using the Ratio Test or the Root Test.

Recall the Ratio Test: For a series \( \sum a_k \), compute \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). If \( L < 1 \), the series converges absolutely; if \( L > 1 \), it diverges; if \( L = 1 \), the test is inconclusive.

Calculate \( \frac{a_{k+1}}{a_k} \) for \( a_k = \frac{2^{k}}{k^{99}} \): \[ \frac{a_{k+1}}{a_k} = \frac{2^{k+1} / (k+1)^{99}}{2^{k} / k^{99}} = 2 \cdot \frac{k^{99}}{(k+1)^{99}} \]

Take the limit as \( k \to \infty \) of \( 2 \cdot \frac{k^{99}}{(k+1)^{99}} \). Since \( \frac{k}{k+1} \to 1 \), this limit simplifies to \( 2 \cdot 1 = 2 \).

Interpret the result: Since the limit \( L = 2 > 1 \), by the Ratio Test, the series \( \sum_{k=1}^{\infty} \frac{2^{k}}{k^{99}} \) diverges.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Ratio Test

The Ratio Test determines the convergence of a series by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.

Root Test

The Root Test analyzes convergence by taking the nth root of the absolute value of the nth term and evaluating its limit as n approaches infinity. If the limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.

Absolute Convergence

A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence guarantees convergence of the original series and is a stronger condition than conditional convergence.

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.∑ (from k = 1 to ∞) (2ᵏ / k⁹⁹)