42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)k! / (eᵏ kᵏ)

검증된 단계별 안내

Identify the series given: \( \sum_{k=1}^{\infty} \frac{k!}{e^{k} k^{k}} \). We want to determine if this infinite series converges or diverges.

Consider using the Ratio Test, which is often effective for series involving factorials and exponential terms. The Ratio Test states that for \( a_k \), if \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \), then the series converges if \( L < 1 \), diverges if \( L > 1 \), and is inconclusive if \( L = 1 \).

Write the ratio \( \frac{a_{k+1}}{a_k} \) explicitly: \[ \frac{a_{k+1}}{a_k} = \frac{\frac{(k+1)!}{e^{k+1} (k+1)^{k+1}}}{\frac{k!}{e^{k} k^{k}}} = \frac{(k+1)!}{e^{k+1} (k+1)^{k+1}} \times \frac{e^{k} k^{k}}{k!} \].

Simplify the expression by canceling factorial terms and exponential terms: \( (k+1)! = (k+1) \times k! \) and \( e^{k+1} = e^{k} \times e \). Substitute these to get a simpler form for the ratio.

After simplification, take the limit as \( k \to \infty \) of the ratio. Use properties of limits and approximations such as \( \left(1 + \frac{1}{k}\right)^k \approx e \) to evaluate the limit and determine whether it is less than, greater than, or equal to 1, which will tell you about the convergence or divergence of the series.

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.

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining whether such a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of series like ∑ k! / (e^k k^k).

Ratio Test

The Ratio Test is a common method to determine 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. This test is particularly useful for series involving factorials and exponentials.

Factorials and Exponential Growth

Factorials (k!) grow very rapidly, but so do exponential functions like e^k and powers like k^k. Comparing the growth rates of these terms helps in applying convergence tests effectively. Recognizing how factorials and exponentials behave is crucial for simplifying terms and evaluating limits in series.

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.∑ (from k = 1 to ∞)k! / (eᵏ kᵏ)