Problem 37

Question

$37-40=$ Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? $$a_{n}=\frac{1}{2 n+3}$$

Step-by-Step Solution

Verified

Answer

The sequence is decreasing and bounded between 0 and $\frac{1}{3}$.

1Step 1: Define the Sequence Behavior

Given the sequence $a_n = \frac{1}{2n + 3}$. To determine if the sequence is increasing, decreasing, or not monotonic, we need to analyze the behavior of $a_n$ as $n$ increases. Observe that as $n$ increases, the denominator $2n + 3$ also increases, causing the value of $a_n$ to decrease because the numerator is constant.

2Step 2: Confirm Decreasing Behavior Mathematically

To mathematically confirm that the sequence is decreasing, consider the terms $a_n$ and $a_{n+1}$. We have:\[ a_n = \frac{1}{2n + 3} \]\[ a_{n+1} = \frac{1}{2(n+1) + 3} = \frac{1}{2n + 5} \]Since $2n + 3 < 2n + 5$, it follows that $\frac{1}{2n+3} > \frac{1}{2n+5}$, thus $a_n > a_{n+1}$. Therefore, the sequence is indeed decreasing.

3Step 3: Determine Boundedness

For boundedness, observe that as $n$ tends to infinity, $2n + 3$ tends to infinity, which causes $a_n = \frac{1}{2n + 3}$ to tend to zero. Thus, the sequence is bounded below by zero. Also, since $a_n$ is a positive fraction, the sequence is bounded above by $\frac{1}{3}$, which is the value of $a_0$. Therefore, the sequence is bounded between 0 and $\frac{1}{3}$.

Key Concepts

Monotonic SequencesBounded SequencesDecreasing Sequences

Monotonic Sequences

A sequence is said to be monotonic if it consistently increases or decreases. Understanding the behavior of a sequence is key to determining its monotonicity. In the case of our sequence, $a_n = \frac{1}{2n + 3}$, we observe how each term changes as $n$ increases. Since the denominator $2n + 3$ gets larger, the fraction itself becomes smaller, indicating a decrease in value. Thus, our sequence is decreasing.

Monotonic sequences can be classified into two types:

Increasing Sequences: where each term is greater than the one before it.
Decreasing Sequences: where each term is less than the one before it, as is the case for $a_n$.

If a sequence alternates directions or has no clear pattern, it's termed non-monotonic.

Bounded Sequences

A sequence is bounded if its values lie within a certain range and do not go to infinity or negative infinity. For our sequence $a_n = \frac{1}{2n + 3}$, we considered what happens as $n$ approaches infinity. The denominator increases, causing $a_n$ to approach zero, the sequence's lower bound.

Being bounded implies two key traits:

Lower Bound: The smallest value that a sequence can approach. In our case, $a_n$ cannot decrease below zero.
Upper Bound: The largest value the sequence can attain. For $a_n$, it is $\frac{1}{3}$, the value when $n = 0$.

By these observations, our sequence lies between 0 and $\frac{1}{3}$, confirming it is bounded.

Decreasing Sequences

Decreasing sequences are a specific type of monotonic sequence where each term is smaller than the one before it. This behavior is clearly seen in $a_n = \frac{1}{2n + 3}$. Here, by comparing successive terms, namely $a_n$ and $a_{n+1}$, we note that $\frac{1}{2n + 3} > \frac{1}{2n + 5}$. This indicates that each term in the sequence is less than its predecessor.

To determine if a sequence is decreasing, we:

Compare consecutive terms to ensure $a_n > a_{n+1}$.
Show mathematically that this pattern holds for all terms within the sequence.

This process verifies that the sequence's value declines over time, characterizing it as a decreasing sequence.

Problem 37

Problem 38

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