Question

Determine whether the following sequences converge or diverge and describe whether they do so monotonically or by oscillation. Give the limit when the sequence converges. $$\left\\{0.2^{n}\right\\}$$

Short answer

If it converges, does it do so monotonically or by oscillation, and what is the limit? Answer: The given sequence \(\left\\{0.2^n\right\\}\) converges monotonically to the limit 0.

Step-by-step solution

Analyzing the sequence

First, we should take a closer look at the form and behavior of the sequence. We are given the sequence: $$\left\\{0.2^n\right\\}$$ The sequence consists of exponential terms with base 0.2, which is a positive real number between 0 and 1. As n increases, these terms get smaller and smaller.

Determine the behavior of the sequence as n approaches infinity

To see the behavior of the sequence when n approaches infinity, we have to examine the limit: $$\lim_{n\to\infty} 0.2^n$$ Since 0.2 is a positive number between 0 and 1, this limit will converge to a value. As n becomes larger and larger, the value of \(0.2^n\) approaches 0.

Determine if the sequence converges or diverges

Based on our analysis in Step 2, the sequence converges. This is because the limit of the sequence exists and is equal to 0: $$\lim_{n\to\infty} 0.2^n = 0$$

Describe whether the sequence converges monotonically or by oscillation

A sequence converges monotonically if its elements either increase or decrease consistently as n approaches infinity. Since the terms in the given sequence continually decrease (as n increases, \(0.2^n\) becomes smaller), the sequence converges monotonically.

Provide the limit of the sequence if it converges

Since we have determined the given sequence converges monotonically, we can provide the limit: $$\lim_{n\to\infty} 0.2^n = 0$$ Thus, the sequence converges monotonically to the limit 0.

Key concepts

Monotonic Convergence

When we discuss monotonic convergence, we are referring to sequences that consistently move towards a limit without fluctuating in direction. A sequence is called monotonic if it is either entirely non-increasing or non-decreasing. In other words, either the elements keep getting smaller or they keep getting larger over time.

Let's apply this to our example sequence, \(\{0.2^n\}\). Given that 0.2 is a number between 0 and 1, when raised to the power of \(n\), each term of this sequence becomes smaller. This consistent decrease means the sequence is monotonic in nature. Since no oscillation occurs (the terms don't jump up and down), the sequence is classified as monotonically decreasing.

• The sequence decreases steadily as each term becomes smaller with an increasing \(n\).
• No fluctuation or oscillation in value occurs.
• This predictable pattern allows us to easily identify monotonic convergence.

Monotonic sequences that are also bounded will converge, making them easier to analyze and predict. For \(\{0.2^n\}\), the approach towards zero is smooth and continuous, assuring us of its monotonic convergence.

Exponential Decay

Exponential decay is a concept that describes how quantities diminish rapidly at rates proportional to their current value. An exponential function of the form \(b^n\) with \(0 < b < 1\) showcases this behavior. Every increase in \(n\) results in the sequence diminishing by a constant factor each time.

In our sequence \(\{0.2^n\}\), each term is "decayed" by multiplying by 0.2, which is less than 1. As a result, each subsequent term is only 20% of its previous value. This shrinking rate illustrates exponential decay, where the values swiftly head towards zero.

• Exponential decay involves repeated multiplication by a shrinking factor.
• This type of sequence rapidly heads towards its limit, often zero.
• It's a common pattern in nature, such as in radioactive decay or cooling processes.

Understanding exponential decay helps recognize and predict patterns in reducing values, which can be particularly significant in applied mathematics.

Sequence Limit

The limit of a sequence is the value that the elements of the sequence approach as the index \(n\) becomes very large. A sequence is said to converge if it approaches a specific limit; if it doesn't approach any specific value, it diverges.

For our sequence \(\{0.2^n\}\), as \(n\) increases, the terms become exceedingly small, trending towards zero. By observing the limit \[ \lim_{n \to \infty} 0.2^n = 0 \] we can see that this sequence not only converges but does so monotonically as well.

• A limit helps identify long-term behavior of a sequence.
• Convergence indicates that as \(n\) gets large, the sequence stabilizes.
• The final limit value provides a conclusion to the sequence's progression.

With the defined limit at zero, it provides assurance that this is what the sequence will eventually settle at, with no oscillations or deviations. Recognizing limits is crucial in mathematics as it assists in understanding complex systems and predicting future states.