6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.

{(−0.7)ⁿ}

검증된 단계별 안내

Identify the given sequence: \(a_n = (-0.7)^n\).

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

Since \(| -0.7 | = 0.7 < 1\), the terms \((-0.7)^n\) get closer to zero as \(n\) increases, so the sequence converges to 0.

Determine the behavior of the sequence: because the base is negative, the terms alternate in sign, causing the sequence to oscillate between positive and negative values.

Conclude that the sequence converges to 0 and oscillates, so it is not monotonic.

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주요 개념

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Sequence Convergence and Divergence

A sequence converges if its terms approach a specific finite value as n approaches infinity; otherwise, it diverges. Determining convergence involves analyzing the behavior of the general term for large n, often using limits.

Monotonicity of Sequences

A sequence is monotonic if it is either entirely non-increasing or non-decreasing. Identifying monotonicity helps understand the sequence's behavior and is useful in proving convergence.

Oscillating Sequences

An oscillating sequence alternates in sign or fluctuates without settling to a single value. Recognizing oscillation is important to distinguish between sequences that converge and those that do not.

6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.{(−0.7)ⁿ}