Question

Describe what it means for a sequence to be bounded.

Short answer

A sequence is bounded if all its terms lie within a fixed interval on the number line.

Step-by-step solution

Introducing the Concept of a Sequence

A sequence is an ordered list of numbers, written as \(a_1, a_2, a_3, \dots\). Each number in the sequence is called a 'term'. For any positive integer \(n\), \(a_n\) represents the 'nth' term in the sequence.

Defining Boundedness

A sequence is said to be bounded if there exists a real number \(M\) such that every term in the sequence is within the interval \([-M, M]\). This means that all the terms \(a_n\) are greater than or equal to \(-M\) and less than or equal to \(M\).

Explanation of Upper and Lower Bounds

A sequence is bounded above if there exists an upper bound \(U\) such that every term \(a_n \leq U\). Similarly, it is bounded below if there exists a lower bound \(L\) such that every term \(a_n \geq L\). A sequence is 'bounded' if it is both bounded above and below.

Visualizing Boundedness

Imagine the sequence terms plotted on a number line. If you can identify two numbers such that the entire sequence is contained between these two numbers, then the sequence is bounded. Think of it as the sequence being 'trapped' within an interval on the number line.

Checking Boundedness with Examples

Consider the sequence \(a_n = \frac{1}{n}\). As \(n\) increases, the terms \(a_n\) decrease, approaching zero but are always positive, thus bounded below by 0. An upper bound for this sequence is 1, as \(\frac{1}{n} \leq 1\) for all \(n \geq 1\). Hence, \(\{a_n\}\) is bounded.

Key concepts

Sequence

A sequence is essentially just an ordered list of numbers. Picture a line of numbers, each with a specific place. Each number in this list is called a 'term'. When working with sequences, we often use a notation like \(a_1, a_2, a_3, \dots\), where \(a_n\) represents the \(n\)-th term of the sequence. This way, we can refer to any term directly by its position.

Sequences can be finite or infinite. A finite sequence has a limited number of terms. An infinite sequence, on the other hand, just keeps going and going. Understanding sequences is critical because they form the foundation for more complex mathematical concepts like series, limits, and calculus.

Whenever you see a sequence, think of each term as part of a family, with each member playing a role depending on its position in the order.

Upper Bound

An upper bound of a sequence is like the ceiling that none of the sequence terms can surpass. Imagine if the terms of your sequence are part of a family living in a house with a low ceiling. If one member grows too tall and hits the ceiling, they have reached the upper bound.

In mathematical terms, if a sequence has an upper bound, there exists a number \(U\) such that every term \(a_n\) is less than or equal to \(U\). This means no matter how far along in the sequence you look, you'll never see a term that's larger than \(U\).

In practice, checking whether a sequence is bounded above is crucial in analysis and helps in understanding the behavior of sequences over large indices.

Lower Bound

A lower bound of a sequence acts as the floor below which no sequence terms can fall. It’s like the floor that keeps all members of our sequence family from sinking downwards.

We say that a sequence is bounded below if there is a number \(L\) such that all terms \(a_n\) are greater than or equal to \(L\). So, even as the sequence progresses, no term will ever dip below this mark.

Knowing about lower bounds helps establish if a sequence behaves consistently and gives stability to the way the sequence unfolds. Lower bounds are particularly useful when analyzing convergence and divergence in sequences.

Number Line

Visualizing a sequence on a number line can make understanding its boundedness much easier. Imagine placing each term of a sequence as a point on a long horizontal line that represents all real numbers.

If there are two specific points on this line that can contain all the terms between them, then the sequence is bounded. These two points symbolize the upper and lower bounds.

The number line helps in visually gauging the spread and density of the terms within a given set of bounds. This can be especially helpful when transitioning to concepts involving continuity and real number properties later in mathematics.

Ordered List of Numbers

At its core, a sequence is an ordered list of numbers, but it’s important to remember that the order is not arbitrary. Each number, or term, has a distinct position, determined by its index, usually denoted as \(n\). This order is crucial because it affects how we interpret the behaviors and properties of the sequence.

For example, in the sequence \(a_n = n^2\), each term is the square of its position: \(a_1 = 1, a_2 = 4\), and so on. The process of forming a sequence is deliberate and follows a specific rule or function.

Sequences are foundational in mathematics, and their ordered nature allows us to explore patterns, determine limits, and solve complex problems by understanding their underlying structures.