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


{(3ⁿ⁺¹ + 3)⁄3ⁿ}

Verified step by step guidance

1. Identify the given sequence: \(a_n = \frac{3^{n+1} + 3}{3^n}\).
2. Rewrite the expression to simplify it by separating the terms in the numerator: \(a_n = \frac{3^{n+1}}{3^n} + \frac{3}{3^n}\).
3. Simplify each term using the properties of exponents: \(\frac{3^{n+1}}{3^n} = 3^{(n+1)-n} = 3^1 = 3\) and \(\frac{3}{3^n} = 3 \cdot 3^{-n} = 3^{1-n}\).
4. Express the sequence as \(a_n = 3 + 3^{1-n}\) and analyze the behavior of \$3^{1-n}\( as \)n$ approaches infinity.
5. Since \$3^{1-n} = \frac{3}{3^n}\( and \)3^n\( grows without bound as \)n \to \infty\(, \)3^{1-n} \to 0\(. Therefore, the limit of the sequence is \)\lim_{n \to \infty} a_n = 3 + 0 = 3$.

Key Concepts

Limits of Sequences

The limit of a sequence is the value that the terms of the sequence approach as the index goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges; otherwise, it diverges. Understanding this helps determine the behavior of sequences like (3ⁿ⁺¹ + 3)/3ⁿ as n grows large.

Properties of Exponents

Exponents follow specific rules such as a^(m+n) = a^m * a^n and a^m / a^n = a^(m-n). Applying these properties simplifies expressions involving powers, which is essential for rewriting and analyzing sequences like (3ⁿ⁺¹ + 3)/3ⁿ to find their limits.

Algebraic Simplification

Algebraic simplification involves rewriting expressions in simpler or more convenient forms. For sequences, this often means factoring or dividing terms to isolate dominant parts, making it easier to evaluate limits. Simplifying (3ⁿ⁺¹ + 3)/3ⁿ helps identify the dominant term as n approaches infinity.