Problem 71 - Calculus: Early Transcendentals

Question

Suppose you know that \( \left\\{ a_n \right\\} \) is a decreasing sequence and all its terms lie between the numbers 5 and 8. Explain why the sequence has a limit. What can you say about the value of the limit?

Step-by-Step Solution

Answer

The sequence has a limit due to the Monotone Convergence Theorem; the limit is at least 5.

Step 1: Understanding Decreasing Sequences

A decreasing sequence is a sequence where each term is less than or equal to the preceding one: \( a_1 \geq a_2 \geq a_3 \geq \ldots \). Given that \( a_n \) is a decreasing sequence, this means it either approaches a specific value or diverges to negative infinity.

Step 2: Identifying Bounded Sequences

The problem states that \( a_n \) lies between 5 and 8 for all terms. This implies the sequence is bounded, meaning \( 5 \leq a_n \leq 8 \) for all \( n \). Being bounded above and below is a key aspect, since it confines the terms of the sequence to a specific interval.

Step 3: Applying the Monotone Convergence Theorem

The Monotone Convergence Theorem tells us that every bounded and monotonic sequence converges. Since \( a_n \) is both decreasing (monotonic) and bounded, the theorem guarantees that the sequence converges to a limit.

Step 4: Determining the Value of the Limit

Because the sequence is decreasing, it will approach a lower bound. Since the sequence is bounded below by 5, this gives a concrete potential limit. Hence the limit of the sequence cannot be less than 5, and it must be the least upper bound (infimum) within the given bounds of the sequence.

Key Concepts

Decreasing Sequences

A decreasing sequence is one where each term is less than or equal to the term before it. This means if we have terms like \(a_1, a_2, a_3, \ldots\), then \(a_1 \geq a_2 \geq a_3 \geq \ldots\). Think about it like descending a staircase, where each step is never higher than the one you are on. In mathematical terms:

• Each term is either the same or smaller than the previous term.
• Decreasing sequences either approach a specific lower bound or could potentially dive infinitely negative.

Decreasing sequences are important in mathematics because they have special behaviors under certain conditions, like being bounded. When combined with the idea of boundedness, they become more predictable in their behavior.

Bounded Sequences

A bounded sequence is one where all its terms are confined within a specified range. For instance, if all terms lie between two numbers, like 5 and 8 in our case, we say the sequence is bounded. Here are some key points about bounded sequences:

• A sequence is bounded above if there is a number larger than or equal to every term in the sequence (like 8 for us).
• It is bounded below if there is a number less than or equal to every term (like 5 in our scenario).
• Being bounded helps ensure that the sequence doesn't go off to infinity in either direction.

The nature of bounded sequences plays a crucial role in determining the behavior of sequences, especially when applying mathematical theorems like the Monotone Convergence Theorem.

Convergence of Sequences

When we talk about the convergence of sequences, we are discussing whether a sequence approaches a specific number as we observe more and more terms. If a sequence converges, it means there is a particular number that the sequence's terms get arbitrarily close to as you consider terms further along in the sequence. Here's what determines convergence:

• A sequence converges if it settles towards a fixed limit. Mathematically, for any small positive number \(\varepsilon\), there is a point after which all terms are within \(\varepsilon\) of the limit.
• The Monotone Convergence Theorem states that if a sequence is both monotonic and bounded, it must converge. This is why our sequence, being both decreasing and bounded, converges.
• For decreasing sequences, the limit they converge to will be the greatest lower bound of all the terms. In our exercise, since the sequence is bounded below by 5, the limit must be no less than 5.

Understanding convergence is key to recognizing how sequences behave as they progress towards infinity.