Problem 66
Question
Determine whether the sequence with th given \(n\) th term is monotonic. Discuss the boundedness of th sequence. Use a graphing utility to confirm your results. \(a_{n}=(-1)^{n}\left(\frac{1}{n}\right)\)
Step-by-Step Solution
Verified Answer
The sequence \(a_{n}=(-1)^{n}\left(\frac{1}{n}\right)\) is not monotonic because it alternates in sign. However, it is bounded as no term in the sequence will have a magnitude greater than 1 or less than -1. The graph of the sequence will confirm this.
1Step 1: Understand the Sequence
The sequence \(a_{n}=(-1)^{n}\left(\frac{1}{n}\right)\) alternates in sign because of the factor \((-1)^n\), and it also decreases in magnitude because \(1/n\) gets smaller as n increases. Thus, it's not monotonic because it does not purely increase or decrease.
2Step 2: Check for Boundedness
To determine the boundedness of the sequence, observe that regardless of the value of n, the maximum magnitude of any term in the sequence is 1 (when n = 1) and the minimum magnitude is 0 (as n approaches infinity). Hence, the sequence is bounded above by 1 and below by -1.
3Step 3: Use Graphing Utility for Verification
Deploy a graphing calculator or software to plot several terms of the sequence. It will clearly show that the sequence alternates sign and shrinks in magnitude, which verifies the results of the previous steps.
Key Concepts
Bounded SequencesAlternating SeriesGraphical Verification
Bounded Sequences
A sequence is bounded if it remains within a fixed range of values. This means there exists a real number that acts as the upper limit, and another as the lower limit, confining all the terms of the sequence within these bounds. For the sequence \(a_{n}=(-1)^{n}\left(\frac{1}{n}\right)\), it is initially important to note that the magnitudes of the terms decrease as \(n\) increases, since \(\frac{1}{n}\) shrinks.
- **Upper Bound**: The maximum value any term can reach in magnitude is 1. This occurs when \(n = 1\) right at the start of the sequence.
- **Lower Bound**: As \(n\) grows larger, the terms move closer to 0, but never actually reach it, implying the magnitude is bounded below by 0.
- **Boundedness**: The sequence is bounded above by 1 and below by -1 (considering both positive and negative values due to alternating signs).
This bounded nature ensures the sequence does not spiral to infinity or decrease infinitely either.
- **Upper Bound**: The maximum value any term can reach in magnitude is 1. This occurs when \(n = 1\) right at the start of the sequence.
- **Lower Bound**: As \(n\) grows larger, the terms move closer to 0, but never actually reach it, implying the magnitude is bounded below by 0.
- **Boundedness**: The sequence is bounded above by 1 and below by -1 (considering both positive and negative values due to alternating signs).
This bounded nature ensures the sequence does not spiral to infinity or decrease infinitely either.
Alternating Series
An alternating series is one where the signs of the sequence terms shift consistently from positive to negative. In the sequence \(a_{n}=(-1)^{n}\left(\frac{1}{n}\right)\), the component \((-1)^n\) is responsible for this alternation of signs.
- When \(n\) is odd, the factor \((-1)^n\) results in a negative sign, making the corresponding term negative.
- When \(n\) is even, it results in a positive sign, thus making the corresponding term positive.
- **Example illustration**: For \(n = 1, 2, 3, 4, ...\), the sequence would be \(-1, 0.5, -0.333, 0.25,...\).
This consistent change from positive to negative showcases the alternating nature of the sequence. It is a clear indication that such a sequence does not maintain a monotonic behavior, as monotonic sequences strictly either increase or decrease.
- When \(n\) is odd, the factor \((-1)^n\) results in a negative sign, making the corresponding term negative.
- When \(n\) is even, it results in a positive sign, thus making the corresponding term positive.
- **Example illustration**: For \(n = 1, 2, 3, 4, ...\), the sequence would be \(-1, 0.5, -0.333, 0.25,...\).
This consistent change from positive to negative showcases the alternating nature of the sequence. It is a clear indication that such a sequence does not maintain a monotonic behavior, as monotonic sequences strictly either increase or decrease.
Graphical Verification
Graphical verification is a practical method to understand the behavior of a sequence visually. By using tools like a graphing calculator or software, you can plot the terms of the sequence and observe their distribution and trends.
When plotting the sequence \(a_{n}=(-1)^{n}\left(\frac{1}{n}\right)\):
When plotting the sequence \(a_{n}=(-1)^{n}\left(\frac{1}{n}\right)\):
- The graph will display the terms alternating along the horizontal axis, swinging from positive to negative values, confirming the alternating sign pattern.
- The magnitude of terms reduces as \(n\) increases, creating a pattern where the points get closer to zero, confirming the decrease in magnitude.
Other exercises in this chapter
Problem 65
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