For what values of r does the sequence {rⁿ} converge? Diverge?

검증된 단계별 안내

Recall that the sequence is given by \(a_n = r^n\), where \(n\) is a natural number and \(r\) is a real number parameter.

To determine convergence or divergence, analyze the behavior of \(r^n\) as \(n\) approaches infinity for different values of \(r\).

Consider the cases based on the absolute value of \(r\): - If \(|r| < 1\), then \(r^n\) approaches 0 as \(n \to \infty\), so the sequence converges to 0. - If \(|r| = 1\), then: * If \(r = 1\), the sequence is constant and converges to 1. * If \(r = -1\), the sequence oscillates between 1 and -1 and does not converge. - If \(|r| > 1\), then \(r^n\) grows without bound in magnitude, so the sequence diverges.

Summarize the results: - Converges to 0 if \(|r| < 1\). - Converges to 1 if \(r = 1\). - Diverges if \(|r| > 1\) or if \(r = -1\) (due to oscillation).

Therefore, the key step is to analyze the absolute value of \(r\) and apply the limit definition of convergence for sequences.

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주요 개념

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Definition of Sequence Convergence

A sequence converges if its terms approach a specific finite limit as n approaches infinity. For the sequence {rⁿ}, this means finding values of r for which rⁿ approaches a finite number when n becomes very large.

Behavior of Exponential Sequences

The sequence {rⁿ} is exponential, where each term is r raised to the power n. Its behavior depends on the magnitude of r: if |r| < 1, the terms get smaller and approach zero; if |r| > 1, the terms grow without bound; if |r| = 1, the sequence may oscillate or remain constant.

Limits Involving Absolute Value

The absolute value of r determines the limit of rⁿ as n approaches infinity. When |r| < 1, rⁿ tends to zero, ensuring convergence. When |r| > 1, rⁿ diverges to infinity or negative infinity. When |r| = 1, the sequence either stays constant (r=1) or oscillates (r=-1), affecting convergence.