Determining Convergence of Sequences
Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = 1 + (0.9)ⁿ

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Identify the general term of the sequence: \(a_n = 1 + (0.9)^n\).

Recall that a sequence converges if its terms approach a finite limit as \(n\) approaches infinity.

Analyze the behavior of the term \((0.9)^n\) as \(n \to \infty\). Since \(0.9\) is between \(-1\) and \(1\), \((0.9)^n\) approaches \(0\).

Use the limit laws to find the limit of \(a_n\): \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(1 + (0.9)^n\right) = 1 + 0\).

Conclude that the sequence converges and its limit is \(1\).

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주요 개념

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Definition of Sequence Convergence

A sequence converges if its terms approach a specific finite value, called the limit, as n approaches infinity. Formally, a sequence {aₙ} converges to L if for every small positive number ε, there exists an N such that for all n > N, |aₙ - L| < ε.

Limits of Exponential Terms

When a sequence includes terms like (r)ⁿ where |r| < 1, these terms approach zero as n becomes very large. This property helps determine the limit of sequences involving exponential decay factors, such as (0.9)ⁿ approaching 0.

Evaluating Limits of Sequences

To find the limit of a sequence, analyze each component separately and use limit laws. For example, in aₙ = 1 + (0.9)ⁿ, since (0.9)ⁿ → 0, the sequence converges to 1 + 0 = 1.

Determining Convergence of SequencesWhich of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.aₙ = 1 + (0.9)ⁿ