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 ∞) ((k / (k + 1)) × 2k²)

검증된 단계별 안내

Identify the general term of the series: \(a_k = \left( \frac{k}{k+1} \right) \times 2^{k^2}\).

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

Compute the ratio \(\frac{a_{k+1}}{a_k} = \frac{\left( \frac{k+1}{k+2} \right) 2^{(k+1)^2}}{\left( \frac{k}{k+1} \right) 2^{k^2}} = \left( \frac{k+1}{k+2} \times \frac{k+1}{k} \right) \times 2^{(k+1)^2 - k^2}\).

Simplify the exponent difference: \((k+1)^2 - k^2 = 2k + 1\), so the ratio becomes \(\left( \frac{(k+1)^2}{k(k+2)} \right) \times 2^{2k+1}\).

Evaluate the limit as \(k \to \infty\) of the ratio. Since \(2^{2k+1}\) grows exponentially, the limit will be infinite, which implies by the Ratio Test that the series 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 an infinite 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 of a series. If the limit of this root as n approaches infinity 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 regardless of term signs, making tests like the Ratio and Root Tests effective tools for determining the behavior of series with positive or negative terms.

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 ∞) ((k / (k + 1)) × 2k²)