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


{2ⁿ⁺¹3⁻ⁿ}

Verified step by step guidance

1. Identify the general term of the sequence, which is given by \(a_n = 2^{n+1} 3^{-n}\).
2. Rewrite the term to express it in a simpler form by separating the powers: \(a_n = 2^{n+1} \times \frac{1}{3^n} = 2 \times 2^n \times 3^{-n}\).
3. Combine the exponential terms with the same exponent: \(a_n = 2 \times \left( \frac{2}{3} \right)^n\).
4. 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 \(n \to \infty\).
5. Conclude that the limit of the sequence \(a_n\) as \(n\) approaches infinity is \$2 \times 0 = 0$.

Key Concepts

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.