Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.


aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)

Verified step by step guidance

1. Identify the dominant terms in the numerator and denominator of the sequence \(a_n = \frac{6^n + 3^n}{6^n + n^{100}}\). Since \$6^n\( and \)3^n\( are exponential functions and \)n^{100}\( is a polynomial, exponential terms will grow faster than polynomial terms as \)n$ approaches infinity.
2. Apply Theorem 10.6, which states that for sequences involving sums of terms with different growth rates, the term with the highest growth rate dominates the behavior of the sequence as \(n \to \infty\).
3. In the numerator, \$6^n\( grows faster than \)3^n\(, so \)6^n\( dominates. In the denominator, \)6^n\( grows faster than \)n^{100}\(, so \)6^n$ dominates there as well.
4. Rewrite the sequence by factoring out \$6^n\( from both numerator and denominator to simplify the expression: \)a_n = \frac{6^n(1 + (\frac{3}{6})^n)}{6^n(1 + \frac{n^{100}}{6^n})}$.
5. Simplify the expression by canceling \$6^n\( and analyze the limits of the remaining terms: \)(\frac{3}{6})^n\( tends to 0 as \)n \to \infty\(, and \)\frac{n^{100}}{6^n}\( also tends to 0. Therefore, the limit of \)a_n\( is the limit of \)\frac{1 + 0}{1 + 0}$.

Key Concepts

Limits of Sequences

The limit of a sequence describes the value that the terms approach as the index goes to infinity. Understanding how to evaluate these limits helps determine whether a sequence converges to a finite number or diverges. Techniques often involve simplifying terms and comparing dominant growth rates.

Dominant Term and Growth Rates

In sequences involving sums of terms with different growth rates, the term with the fastest growth dominates the behavior as n becomes large. Comparing exponential terms like 6ⁿ and 3ⁿ or polynomial terms like n¹⁰⁰ helps identify which terms control the limit.

Theorem 10.6 (Limit of Ratio of Sequences)

Theorem 10.6 typically refers to a result about limits of sequences involving ratios, stating that if the dominant terms in numerator and denominator grow at the same rate, the limit is the ratio of their leading coefficients. This theorem simplifies finding limits of complex sequences by focusing on dominant terms.