23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 1 to ∞) k / eᵏ

검증된 단계별 안내

Identify the series given: \( \sum_{k=1}^{\infty} \frac{k}{e^{k}} \). Notice that the terms involve \( k \) in the numerator and an exponential \( e^{k} \) in the denominator.

Consider the behavior of the terms \( a_k = \frac{k}{e^{k}} \) as \( k \to \infty \). Since the denominator grows exponentially and the numerator grows linearly, the terms \( a_k \) approach zero, which is a necessary condition for convergence.

Apply the Divergence Test first: check if \( \lim_{k \to \infty} a_k \neq 0 \). If the limit is not zero, the series diverges. Here, the limit is zero, so the Divergence Test is inconclusive.

Use the Integral Test or compare with a known convergent series. Since \( e^{k} \) grows faster than any polynomial, compare \( \frac{k}{e^{k}} \) to \( \frac{1}{e^{k/2}} \) or recognize it as a series with terms decreasing exponentially.

Conclude that the series converges by comparison to a convergent geometric series or by applying the Integral Test to the function \( f(x) = \frac{x}{e^{x}} \), which is positive, continuous, and decreasing for \( x \geq 1 \).

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.

Divergence Test

The Divergence Test states that if the limit of the terms of a series does not approach zero, the series diverges. It is a quick initial check to determine if a series cannot converge, but if the limit is zero, the test is inconclusive.

Integral Test

The Integral Test relates the convergence of a series to the convergence of an improper integral of a related function. If the integral of the continuous, positive, decreasing function from which the series terms are derived converges, then the series converges as well.

p-series Test

The p-series Test applies to series of the form ∑ 1/n^p. Such a series converges if and only if p > 1, and diverges otherwise. It is useful for comparing or identifying the behavior of series with terms involving powers of n.

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.∑ (k = 1 to ∞) k / eᵏ