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}

Verified step by step guidance

1. Identify the given sequence: \(\{1.00001^n\}\), where \(n\) is a positive integer increasing without bound.
2. 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.
3. 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.
4. 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.
5. Summarize: the sequence diverges (does not converge to a finite limit) and is monotonic increasing; it does not oscillate.

Key Concepts

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.