Question

Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit. $$ a_{n}=\frac{\ln (\cos n)}{n} $$

Short answer

The sequence is not bounded, not monotone, and is divergent.

Step-by-step solution

Analyze Boundedness

To determine if the sequence \( a_n = \frac{\ln(\cos n)}{n} \) is bounded, consider the range of \( \cos n \). Since \( \cos n \) oscillates between -1 and 1, and the natural logarithm is undefined for non-positive values, \( \ln(\cos n) \) is only defined when \( \cos n \) is positive, i.e., between 0 and 1. However, since \( \cos n \) also includes negative values and zero, the sequence is not always defined, indicating that it is not bounded in the traditional sense.

Analyze Monotonicity

Check if the sequence is monotone by analyzing the behavior of \( a_n \). The values of \( \cos n \) lead to \( \ln(\cos n) \) frequently changing sign and magnitude, implying \( a_n \) is not consistently increasing or decreasing. Therefore, the sequence is not monotone.

Evaluate Convergence

Explore if the sequence converges by examining its limit as \( n \to \infty \). Since \( \cos n \) oscillates between -1 and 1, \( \ln(\cos n) \) is undefined or negative. With \( a_n = \frac{\ln(\cos n)}{n} \) potentially undefined for infinitely many terms, it lacks convergence in a traditional sense. Additionally, the undefined values disrupt the sequence's potential convergence.

Conclude on Convergence and Divergence

Given the oscillating behavior of \( \cos n\) leading to both undefined and negative \( \ln(\cos n) \), the sequence \( a_n \) does not converge. Since the sequence lacks a limit and varies significantly, it is divergent.

Key concepts

Boundedness

A sequence is termed as bounded if all its terms lie within a certain fixed interval on the number line. For the sequence \( a_n = \frac{\ln(\cos n)}{n} \), we need to consider the behavior of \( \cos n \). Since \( \cos n \) oscillates between -1 and 1, the natural logarithm \( \ln(\cos n) \) is only defined for positive values of \( \cos n \). This limitation makes the sequence problematic, as \( \cos n \) frequently dips into negative territory and reaches zero, where \( \ln(\cos n) \) is undefined. This issue shows that the sequence is not strictly bounded within a range because at many points, \( \ln(\cos n) \) is not even defined. Thus, without a consistent boundary for all terms, we conclude that the sequence is not bounded.

• Bounded sequences have limits for their terms.
• The oscillation and undefined nature of \( \ln(\cos n) \) prevents boundedness.

Monotonicity

Monotonic sequences are those that either consistently increase or decrease. To determine this, consider the sequence \( a_n = \frac{\ln(\cos n)}{n} \). For a monotonic series, the sequence terms must follow a straight path without alternating. However, the periodic nature of \( \cos n \) leads \( \ln(\cos n) \) to frequently change in sign and magnitude. The division by \( n \) further complicates matters as it alters the magnitude of \( a_n \) across different terms. These fluctuations signal that the sequence does not follow a smooth monotonic path, either upward or downward. Consequently, the sequence does not exhibit monotonicity.

• Monotonicity requires consistent directional behavior in a sequence.
• The fluctuating \( \ln(\cos n) \) values result in no monotone pattern.

Convergence and Divergence

Convergence in a sequence means it approaches a specific value as the term number \( n \) increases to infinity. Let's analyze the sequence \( a_n = \frac{\ln(\cos n)}{n} \). The oscillation of \( \cos n \) affects \( \ln(\cos n) \), introducing negative values and often making terms undefined due to the requirements of the logarithm. As we progress to larger \( n \), these undefined or oscillatory sections disrupt any hope of approaching a single limit value. Because the sequence cannot maintain a trend towards a particular number, it is considered divergent. Divergence implies there is no single value that the sequence tends to over time. Thus, the sequence \( a_n \) is divergent.

• Convergence requires stability towards a fixed point.
• Here, undefined behaviors and fluctuations prevent convergence.
• A divergent sequence shows inconsistency and lack of limit.