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

{2ⁿ⁺¹3⁻ⁿ}

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Identify the general term of the sequence, which is given by \(a_n = 2^{n+1} 3^{-n}\).

Rewrite the term to express it in a simpler form by separating the powers: \(a_n = 2^{n+1} \times \frac{1}{3^n} = 2 \times 2^n \times 3^{-n}\).

Combine the exponential terms with the same exponent: \(a_n = 2 \times \left( \frac{2}{3} \right)^n\).

Analyze the base of the exponential term \(\left( \frac{2}{3} \right)\) to determine the behavior as \(n\) approaches infinity. Since \(\frac{2}{3} < 1\), the term \(\left( \frac{2}{3} \right)^n\) approaches 0 as \(n \to \infty\).

Conclude that the limit of the sequence \(a_n\) as \(n\) approaches infinity is \(2 \times 0 = 0\).

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주요 개념

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

Limits of Sequences

The limit of a sequence describes the value that the terms of the sequence approach as the index goes to infinity. If the terms get arbitrarily close to a finite number, the sequence converges; otherwise, it diverges.

Exponential Functions and Their Behavior

Sequences involving exponential terms like 2^(n+1) and 3^(-n) grow or decay depending on the base. Understanding how exponential growth and decay affect the sequence's terms is crucial to determining the limit.

Simplifying and Manipulating Sequences

To find limits, it is often necessary to rewrite the sequence in a simpler form, such as combining exponents or factoring terms. This helps reveal dominant terms and makes it easier to analyze the behavior as n approaches infinity.

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