13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.

{(1 + (2 / n))ⁿ}

검증된 단계별 안내

Identify the given sequence as \( a_n = \left(1 + \frac{2}{n}\right)^n \).

Recall the important limit definition related to sequences of the form \( \left(1 + \frac{x}{n}\right)^n \), which approaches \( e^x \) as \( n \to \infty \).

In this problem, recognize that \( x = 2 \), so the sequence resembles \( \left(1 + \frac{2}{n}\right)^n \).

Apply the limit property: \( \lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^n = e^2 \).

Conclude that the sequence converges and its limit is \( e^2 \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index n becomes very large. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.

Exponential Limit Form (1 + 1/n)^n

The expression (1 + 1/n)^n is a classic limit that approaches the mathematical constant e as n approaches infinity. Variations like (1 + k/n)^n, where k is a constant, approach e raised to the power k, which helps evaluate similar limits.

Properties of the Number e

The number e (~2.71828) is the base of natural logarithms and arises naturally in limits involving growth processes. It is defined as the limit of (1 + 1/n)^n as n approaches infinity and is fundamental in calculus and exponential functions.

13–52. Limits of sequencesFind the limit of the following sequences or determine that the sequence diverges.{(1 + (2 / n))ⁿ}