Problem 2 - Precalculus Mathematics for Calculus

Question

A sequence \(a_{1}, a_{2}, a_{3}, \ldots\) has the limit \(L\) if the \(n\) th term \(a_{n}\) of the sequence can be made arbitrarily close to _____ by taking \(n\) to be sufficiently _____, If the limit exists, we say that the sequence __________; otherwise, the sequence __________.

Step-by-Step Solution

Answer

The sequence converges if it approaches a limit 'L' as 'n' becomes large.

Step 1: Identifying Key Components

Start by identifying the key terms: 'sequence', 'limit', 'arbitrarily close', and 'sufficiently'. These terms relate to the definition of a sequence converging to a limit.

Step 2: Understanding Sequence Convergence

A sequence converges to a limit if the terms of the sequence get closer and closer to a specific value, known as the limit, as the sequence progresses.

Step 3: Filling in the Blanks

The limit in question is represented by the letter 'L'. To say a term approaches this limit, you describe it as 'arbitrarily close to L'. To express that this happens as the sequence goes on, you say 'by taking n to be sufficiently large'.

Step 4: Defining Convergence and Divergence

If the limit exists, the sequence is said to 'converge'. If the limit does not exist, the sequence is said to 'diverge'.

Key Concepts

Limit of a Sequence

In mathematics, a sequence is a collection of numbers arranged in a specific order. The concept of the "limit of a sequence" is fundamental when discussing sequences. Simply put, the limit is the value that the terms of a sequence get closer to, as we progress through the sequence. If the terms of a sequence get arbitrarily close to a particular number as their index becomes very large, we say that the sequence approaches this number. This particular number is what we call the "limit" of the sequence.

• The notation typically used is: \( \lim_{{n \to \infty}} a_n = L \). This means as \( n \) (the term position) goes to infinity, \( a_n \) (the term itself) gets closer to \( L \) (the limit).
• The terms "arbitrarily close" and "sufficiently large" are essential. "Arbitrarily close" means friends can make the distance between sequence terms and the limit as small as they like. "Sufficiently large" means that there is a threshold beyond which all terms satisfy this closeness condition.

Understanding the limit of a sequence is essential for discussing whether a sequence is convergent or divergent.

Convergent Sequence

A sequence is labeled as "convergent" when its terms approach a specific value, known as the limit, as the sequence progresses. It’s like an improving archer consistently getting closer to the center of a target with each shot.
This behavior means, for every small margin of error you select, no matter how tiny (epsilon \( \epsilon \)), there exists a point beyond which all subsequent terms in the sequence are within that margin from the limit.

• Mathematically, we express this as: for every \( \epsilon > 0 \), there exists a number \( N \), such that for all \( n > N \), the difference \( |a_n - L| < \epsilon \).
• This mathematical definition ensures that the terms are not only getting closer but also remain near the limit as they continue to progress in the sequence.

Having a convergent sequence helps us make predictions and calculations relating to the long-term behavior of a sequence.

Divergent Sequence

When a sequence does not have a limit, it is referred to as a "divergent" sequence. Unlike convergent sequences, divergent sequences do not settle down near a particular value. This lack of a limit can be due to the terms increasing or decreasing without bound, or because they oscillate without ever settling.

• For a sequence to be divergent, it essentially means it violates the criteria for convergence. This means there is no single number that every term gets arbitrarily close to.
• This behavior can be seen in sequences where the terms keep growing, for example, like the sequence of natural numbers \( 1, 2, 3, \ldots \), or where they oscillate, like \( (-1)^n \), which bounces between \( 1 \) and \( -1 \).

Understanding divergence is critical because it tells us that the sequence does not stabilize, which can influence how mathematicians approach problems involving such sequences.