Question

Growth rates of sequencesUse Theorem 10.6 to find the limit of the following sequences or state that they diverge.aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)

Accepted Answer

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. 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 $$n 	o \infty. In $$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. 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}$$}{6^n})}. $$Simplify the expression by canceling $$6^n$$ and analyze the limits of the remaining terms: $$(\frac{3}{6})^n$$ tends to 0 as $$n 	o \infty$$, and $$\frac{n$$^{100}$$}{6^n}$$ also tends to 0. Therefore, the limit of $$a_n is $$the limit of $$\frac{1 + 0}{1 + 0}.$$

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)

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Key Concepts

Limits of Sequences

Dominant Term and Growth Rates

Theorem 10.6 (Limit of Ratio of Sequences)