Question

Find \(\overline{\lim } x_{n}\) and \(\lim x_{n}\) if (a) \(x_{n}=c\) (constant); (b) \(x_{n}=-n ;\) (c) \(x_{n}=n ;\) and (d) \(x_{n}=(-1)^{n} n-n .\) Does \(\lim x_{n}\) exist in each case?

Short answer

(a) Yes, limit is \(c\); (b) No, \(\overline{\lim} = -\infty\); (c) No, \(\overline{\lim} = \infty\); (d) No, \(\overline{\lim} = 0\).

Step-by-step solution

Understanding Lim Sup and Lim

The limit superior, denoted as \(\overline{\lim } x_n\), is defined as the largest limit point of a sequence. The limit of a sequence, \(\lim x_n\), is the value the sequence approaches as \(n\) becomes very large. If \(\lim x_n\) exists, it would be equal to \(\overline{\lim } x_n\) when the sequence converges.

Case (a): Sequence is Constant

For \(x_n = c\), where \(c\) is a constant, the sequence is constant for all \(n\). Hence, both \(\overline{\lim } x_n\) and \(\lim x_n\) are equal to \(c\). The limit exists and is \(c\).

Case (b): Sequence \(x_n = -n\)

Here, the sequence \(x_n = -n\) decreases without bound. Thus, \(\overline{\lim } x_n = -\infty\) since the sequence becomes infinitely negative as \(n\) increases. The limit of the sequence also does not exist, so \(\lim x_n = -\infty\).

Case (c): Sequence \(x_n = n\)

The sequence \(x_n = n\) increases without bound. Therefore, \(\overline{\lim } x_n = \infty\), and the limit does not exist as there is no finite value the sequence approaches. Hence, \(\lim x_n\) does not exist.

Case (d): Sequence \(x_n = (-1)^n n - n \)

This sequence can be rewritten as \((-1)^n n - n = (-1)^n n + (-n) = (1-2(-1)^n) n\). For even \(n\), the sequence becomes \(0\), and for odd \(n\), it becomes \(-2n\). Thus, \(\overline{\lim } x_n = 0\) because 0 reoccurs infinitely often, making it the largest limit point. The sequence does not converge to any single value, so \(\lim x_n\) does not exist.

Key concepts

Convergent Sequences

In mathematics, a convergent sequence is a sequence whose terms approach a specific value, called the limit, as the sequence progresses. Imagine a sequence as a list of numbers that keep moving closer to a particular number, like bees flying closer to a flower. For example, consider the sequence where each term is the number '2', i.e., \( x_n = 2 \). This sequence is constant and every term is equal to '2'.
This means that as you go further in the sequence, all values stay the same. Thus, the sequence converges to the limit 2.

• If a sequence converges, both the limit superior and the limit are the same.
• In practice, the main characteristic of a convergent sequence is that it does not wobble around but instead settles at a single number.

When dealing with sequences, understanding convergence is crucial for determining their behavior at infinity.

Divergent Sequences

Divergent sequences are sequences that do not settle at any particular value as they progress. If you imagine trying to follow these sequences, they would leave you dizzy as they never stay in one place. Take, for instance, the sequence \( x_n = n \), where each term just keeps increasing: 1, 2, 3, and so on.
Here, there’s clearly no one value that these terms settle on, and as such, the sequence is said to diverge.

• These sequences often tend towards infinity or negative infinity.
• A divergent sequence does not have a finite limit.
• In the sequence \( x_n = -n \), the terms go towards negative infinity.

When a sequence is divergent, the terms become too large or too small, and it's impossible to predict them landing on one specific number.

Bounded Sequences

Bounded sequences are those sequences that stay within a fixed interval and do not stray beyond it. Visualize these as birds flying in a cage—no matter how far they fly, they cannot escape the cage boundaries. For instance, consider a sequence \( x_n = (-1)^n \).
This sequence oscillates between -1 and 1, never surpassing these bounds. Such sequences may be bounded above, bounded below, or both.

• A sequence is bounded above if there is a number greater than or equal to every term in the sequence.
• It is bounded below if there is a number smaller than or equal to every term.
• If both conditions are satisfied, the sequence is simply bounded.

Understanding whether a sequence is bounded helps in analyzing and managing the long-term behavior of the sequence.

Limit Points

Limit points are fundamental to understanding the nature of sequences. A limit point of a sequence is a number that the sequence can get arbitrarily close to infinitely often. Think of this as traffic converging towards a bustling city square.
For example, in the sequence \( x_n = (-1)^n n - n \), the number '0' is a limit point because the sequence reaches '0' an infinite number of times when 'n' is even.

• Limit points may be actual limits if a sequence converges.
• If the sequence doesn’t converge, the largest limit point can be identified as the limit superior (\( \overline{\lim} x_n \)).
• Discovering limit points is often like sleuthing out repeating patterns within a sequence.

Having clarity on limit points is essential when defining whether a sequence converges to a specific value or behaves unpredictably.