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. $$\left\\{(-0.7)^{n}\right\\}$$

Short answer

If it converges, find the limit and describe if it converges monotonically or by oscillation. Answer: The sequence \((-0.7)^n\) converges with a limit of 0 and converges by oscillation.

Step-by-step solution

Identify the common ratio r and sequence type

The sequence is given by \((-0.7)^{n}\). We can see that it is a geometric sequence with the common ratio \(r=(-0.7)\). The sequence is alternating because it has a negative common ratio.

Determine if the sequence converges

A geometric sequence converges if the absolute value of the common ratio r is less than 1. In this case, \(|-0.7|=0.7 < 1\). Since the absolute value of r is less than 1, the sequence converges.

Find the limit of the sequence

As the sequence converges, we can find the limit L as \(n \to \infty\). For a converging geometric sequence, the limit is given by \(L = \lim_{n \to \infty} r^n\). In this case, we find the limit by taking \(\lim_{n \to \infty} (-0.7)^n\). Since \(|-0.7| < 1\), the limit must be 0.

Determine the type of convergence

Now, we need to determine if the sequence converges monotonically or by oscillation. Since the sequence has a negative common ratio, it oscillates between positive and negative values. Thus, the sequence converges by oscillation. To summarize the solution, 1. The sequence \((-0.7)^n\) converges. 2. The limit of the sequence is 0. 3. The sequence converges by oscillation.

Key concepts

Geometric Sequence

Geometric sequences are sequences of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This type of sequence is expressed in the form \( a, ar, ar^2, ar^3, \ldots \), where \( a \) is the initial term and \( r \) is the common ratio.

Key aspects of geometric sequences include:

• If the absolute value of the common ratio \(|r| < 1\), the sequence converges.
• If \(|r| \geq 1\), the sequence diverges.

For example, the sequence \((-0.7)^n\) is a geometric sequence with \( r = -0.7 \). Because the absolute value \( |r| = 0.7 < 1 \), this tells us immediately that the sequence converges.

Understanding geometric sequences help in predicting the behavior of series and can also be applied in fields such as finance, physics, and computer science.

Alternating Sequence

An alternating sequence is a sequence in which the signs of the terms alternate between positive and negative. In many cases, this alteration can be attributed to a negative common ratio. This is evident in our sequence, \((-0.7)^n\), where the negative common ratio \(-0.7\) causes the terms to switch signs with each iteration:

• When \(n\) is even, \((-0.7)^n\) results in a positive term.
• When \(n\) is odd, \((-0.7)^n\) results in a negative term.

Alternating sequences can either converge or diverge based on additional properties like the absolute value of the common ratio. When evaluating these sequences, it's essential to recognize that alternating signs alone do not determine convergence, as convergence is dependent on other factors such as whether the sequence is a geometric sequence and \(|r| < 1\).

Understanding alternating sequences is crucial in determining the behavior of terms within a sequence and further analyzing their convergence or divergence properties.

Limit of a Sequence

The limit of a sequence refers to the value that the terms of the sequence approach as the index \(n\) goes to infinity. For sequences where terms become infinitely large or small, finding the limit helps in identifying whether the sequence converges or diverges. The limit does not exist if the sequence diverges.

For a converging geometric sequence like \((-0.7)^n\), the limit can be determined by observing that as \(n\) becomes very large, \(|-0.7|^n\) becomes very small. This implies that the limit of the sequence at \(n \to \infty\) is 0, given the absolute value is less than one.

Key points about the limit of sequences include:

• If the limit exists and is finite, the sequence converges.
• If the limit does not exist or is infinite, the sequence diverges.

Understanding limits is essential in various branches of mathematics and other scientific disciplines, as it allows for precise descriptions of the behavior of functions and sequences.

Convergence by Oscillation

Convergence by oscillation occurs when the terms of a sequence converge to a certain value, but the terms oscillate between positive and negative values along the way. This type of convergence is often associated with sequences that have alternating signs, like our example \((-0.7)^n\).

Oscillatory convergence is different from monotone convergence where a sequence approaches its limit by settling into a consistent increasing or decreasing pattern without switching signs. In oscillatory convergence:

• The sequence approaches the limit, but the path includes alternating terms.
• The sequence \((-0.7)^n\) converges to 0, but because of the negative common ratio, it does so by oscillating between negative and positive values.

Recognizing when a sequence converges by oscillation aids in understanding diverse types of sequence behaviors and helps in calculating precise limits amid these oscillations. This concept has applications in oscillatory phenomena observed in physics and engineering.