Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = 8^(1/n)

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Identify the given sequence: \(a_n = 8^{\frac{1}{n}}\).

Recall that as \(n\) approaches infinity, the term \(\frac{1}{n}\) approaches 0.

Rewrite the sequence in terms of an exponential function: \(a_n = e^{\ln(8^{\frac{1}{n}})} = e^{\frac{1}{n} \ln(8)}\).

Analyze the limit of the exponent as \(n \to \infty\): \(\lim_{n \to \infty} \frac{1}{n} \ln(8) = 0\) because \(\ln(8)\) is a constant and \(\frac{1}{n} \to 0\).

Use the continuity of the exponential function to find the limit of the sequence: \(\lim_{n \to \infty} a_n = e^0 = 1\). Therefore, the sequence converges to 1.

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Sequence and Limit

A sequence is an ordered list of numbers defined by a formula for its nth term. The limit of a sequence is the value that the terms approach as n becomes very large. Understanding how to find and interpret limits is essential to determine if a sequence converges or diverges.

Convergence and Divergence of Sequences

A sequence converges if its terms approach a finite number as n approaches infinity; otherwise, it diverges. Recognizing whether a sequence converges or diverges helps in analyzing its long-term behavior and is fundamental in calculus.

Properties of Exponential and Root Functions

Expressions like 8^(1/n) involve roots and exponents. As n increases, the nth root of a positive number approaches 1. Understanding these properties allows you to evaluate the limit of sequences involving roots and powers effectively.

Convergence and DivergenceWhich of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.aₙ = 8^(1/n)