12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (–1)ⁿ / 0.9ⁿ

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Identify the general term of the sequence: \(a_n = \frac{(-1)^n}{0.9^n}\).

Rewrite the denominator to understand its behavior as \(n\) increases: since \(0.9^n = (\frac{9}{10})^n\), it is a positive number less than 1 raised to the power \(n\).

Recall that when a number between 0 and 1 is raised to larger and larger powers, it approaches 0, so \(0.9^n \to 0\) as \(n \to \infty\).

Consider the numerator \((-1)^n\), which oscillates between \(+1\) and \(-1\) depending on whether \(n\) is even or odd.

Combine these observations: since the denominator approaches 0 and the numerator oscillates between \(+1\) and \(-1\), analyze the behavior of the fraction \(\frac{(-1)^n}{0.9^n}\) as \(n\) grows large to determine if the limit exists or diverges.

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주요 개념

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Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index n goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.

Behavior of Exponential Functions

Exponential functions with bases greater than 1 grow without bound, while those with bases between 0 and 1 approach zero as n increases. Understanding this helps determine the dominant behavior of terms like 0.9ⁿ in sequences.

Alternating Sequences

An alternating sequence changes sign with each term, often expressed as (–1)ⁿ or (–1)ⁿ⁺¹. Such sequences may oscillate and not have a limit unless the magnitude of terms approaches zero.

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.aₙ = (–1)ⁿ / 0.9ⁿ