Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
∑ (from n = 1 to ∞) (1 − 1/n)ⁿ

검증된 단계별 안내

First, identify the general term of the series: \(a_n = \left(1 - \frac{1}{n}\right)^n\).

Next, analyze the behavior of the term \(a_n\) as \(n\) approaches infinity by finding the limit \(\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n\).

Recall the known limit \(\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n = e^{-1}\), which is a nonzero constant.

Since the terms \(a_n\) do not approach zero, apply the Divergence Test (also called the nth-term test for divergence), which states that if \(\lim_{n \to \infty} a_n \neq 0\), then the series \(\sum a_n\) diverges.

Conclude that the series \(\sum_{n=1}^\infty \left(1 - \frac{1}{n}\right)^n\) diverges because its terms do not tend to zero, so it does not converge and therefore does not have a finite sum.

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주요 개념

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

Convergence and Divergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit; otherwise, it diverges. Determining convergence involves testing whether the terms decrease sufficiently fast and if the sum stabilizes as more terms are added.

Limit of the General Term

A necessary condition for series convergence is that the general term approaches zero as n approaches infinity. If the limit of the term (1 - 1/n)^n is not zero, the series must diverge by the Test for Divergence.

Behavior of the Term (1 - 1/n)^n

The expression (1 - 1/n)^n approaches 1/e as n becomes large. Understanding this limit helps determine the behavior of the series terms and whether the series can converge or diverge.

Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.∑ (from n = 1 to ∞) (1 − 1/n)ⁿ