Problem 1 - Calculus: Early Transcendentals

Question

(a) What is a sequence? (b) What does it mean to say that \( \lim_{n \to \infty} a_n = 8? \) (c) What does it mean to say that \( \lim_{n \to \infty} a_n = \infty? \)

Step-by-Step Solution

Answer

A sequence is a list of numbers. \( \lim_{n \to \infty} a_n = 8 \) means terms approach 8 as \( n \) increases. \( \lim_{n \to \infty} a_n = \infty \) means terms become arbitrarily large.

Step 1: Understanding a Sequence

A sequence is an ordered list of numbers arranged in a specific pattern or order. Each number in this list is called a term of the sequence. For example, the sequence \( \{1, 2, 3, 4, \ldots \} \) is an arithmetic sequence where each term increases by 1.

Step 2: Explaining the Limit of a Sequence

When we say that \( \lim_{n \to \infty} a_n = 8 \), it means that as the term number \( n \) becomes very large, the terms \( a_n \) of the sequence approach the value 8. In other words, for any small positive number \( \epsilon \), there exists a natural number \( N \) such that for all \( n > N \), the absolute difference \( |a_n - 8| < \epsilon \).

Step 3: Explaining the Sequence Tending to Infinity

When \( \lim_{n \to \infty} a_n = \infty \), it means that as the term number \( n \) increases, the terms \( a_n \) of the sequence become arbitrarily large. In formal terms, for any large number \( M \), there exists a natural number \( N \) such that for all \( n > N \), the terms \( a_n > M \).

Key Concepts

Limit of a Sequence

A sequence is essentially a list of numbers following a particular rule. When discussing the limit of a sequence, we explore the behavior of its terms as the sequence progresses towards infinity. The term "limit" refers to a value that the terms of a sequence tend to get closer to, as the term number increases. For instance, if we say that \( \lim_{n \to \infty} a_n = 8 \), this signifies that as "n" grows larger, the sequence terms \( a_n \) get closer and closer to the value 8.
Understanding this concept can be quite essential:

• If you pick any tiny positive number, say \( \epsilon \), regardless of how small, the terms of the sequence will get within this distance from 8 as "n" becomes very large.
• In mathematical terms, for every \( \epsilon > 0 \), there exists a natural number \( N \) such that for all \( n > N \), the absolute difference \( |a_n - 8| < \epsilon \).

The idea of a limit helps in understanding the eventual behavior of sequences, which can be essential in calculus and analysis.

Convergence of a Sequence

The convergence of a sequence is closely tied to the concept of a limit. When a sequence approaches a specific limit as "n" increases, we say the sequence is converging. A sequence is said to converge if it progressively approaches a certain number, making no drastic deviations over time.
Convergence can be described as:

• The terms of the sequence will get indefinitely closer to the limit as "n" becomes larger.
• No matter how much the sequence wiggles at the start, by choosing a sufficiently large "n", the distance from the limit to \( a_n \) can be made arbitrarily small.
• In essence, a convergent sequence "settles down" to a limit.

This foundational concept is pivotal, as it ensures the predictability and stability of a sequence's behavior over time.

Divergence of a Sequence

Divergence of a sequence refers to when a sequence does not converge to any finite limit. This occurs when the terms of the sequence either increase indefinitely or oscillate without approaching a fixed number as "n" becomes very large.
Key aspects of divergence include:

• If we say that \( \lim_{n \to \infty} a_n = \infty \), it implies that as "n" increases, the terms \( a_n \) grow larger and larger, never settling down to a finite number.
• For any large threshold \( M \), there exists a point in the sequence after which all terms are greater than this threshold.
• In broader terms, divergent sequences do not tend towards any specific value, indicating no single limit is approached.

Understanding divergence helps us recognize when a sequence's behavior is unbounded or erratic, which is crucial for distinguishing between sequences with and without clear limits.