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.

{1.00001ⁿ}

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Identify the given sequence: $\{1.00001^n\}$, where $n$ is a positive integer increasing without bound.

Recall that for sequences of the form $a^n$, the behavior depends on the base $a$: if $|a| > 1$, the sequence grows without bound; if $|a| = 1$, it is constant or oscillates; if $|a| < 1$, it converges to zero.

Since $1.00001$ is slightly greater than 1, the sequence \$1.00001^n$ will increase as $n$ increases, tending towards infinity, so it does not converge to a finite limit.

Determine monotonicity: because the base is greater than 1 and positive, each term is larger than the previous one, so the sequence is strictly increasing and monotonic.

Summarize: the sequence diverges (does not converge to a finite limit) and is monotonic increasing; it does not oscillate.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sequence Convergence and Divergence

A sequence converges if its terms approach a specific finite limit as n approaches infinity. If the terms do not approach any finite value, the sequence diverges. Determining convergence involves analyzing the behavior of the general term for large n.

Monotonicity of Sequences

A sequence is monotonic if it is either entirely non-increasing or non-decreasing. Monotonic sequences have terms that consistently move in one direction, which helps in understanding their long-term behavior and potential convergence.

Exponential Sequences and Limits

Sequences of the form a^n, where a is a constant, exhibit different behaviors depending on the value of a. If |a| > 1, the sequence grows without bound (diverges); if |a| < 1, it converges to zero; if a = 1, it is constant. This concept is key to analyzing the given sequence 1.00001^n.

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.{1.00001ⁿ}