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Chapter 9: Problem 14

Determine whether the statement is true or false. Explain your answer. A sequence \(\left\\{a_{n}\right\\}\) is monotone if \(a_{n+1}-a_{n} \neq 0\) for all \(n \geq 1\).

Short Answer

Expert verified
The statement is false; the condition does not guarantee monotonicity.

Step by step solution

01

Definition of a Monotone Sequence

A sequence \( \{a_n\} \) is called monotone if it is either entirely non-increasing or non-decreasing. This means \( a_{n+1} \geq a_n \) for all \( n \) if the sequence is non-decreasing, or \( a_{n+1} \leq a_n \) for all \( n \) if the sequence is non-increasing.
02

Analyze the Given Condition

The condition given is \( a_{n+1} - a_n eq 0 \). This implies that for each \( n \), \( a_{n+1} \) is not equal to \( a_n \). However, this condition does not specify whether \( a_{n+1} \) should be strictly greater than or strictly less than \( a_n \), only that they are not equal.
03

Relation between Condition and Monotonicity

A sequence having \( a_{n+1} - a_n eq 0 \) does not ensure monotonicity. For example, alternating numbers like 1, 2, 1, 2,... satisfy \( a_{n+1} - a_n eq 0 \) but aren't monotone. Monotonicity requires a consistent direction (non-increasing or non-decreasing), not just inequality.
04

Conclusion about the Given Statement

The statement is false. The condition \( a_{n+1} - a_n eq 0 \) by itself does not ensure that the sequence is monotone because it does not restrict the changes to be in one consistent direction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Increasing sequence
An increasing sequence is a type of monotone sequence where each term is equal to or greater than the previous term. In mathematical terms, a sequence \( \{a_n\} \) is considered non-decreasing if for every index \( n \), the inequality \( a_{n+1} \geq a_n \) holds true.
When a sequence is strictly increasing, it means that each term is greater than the previous one, denoted as \( a_{n+1} > a_n \).
  • Example: The sequence 2, 4, 6, 8,... is strictly increasing since each term increases by 2.
To determine if a sequence is increasing, one must check this inequality condition for all terms in the sequence. This consistent upward trend ensures the sequence is non-decreasing or monotonic in an increasing sense.
Decreasing sequence
Conversely, a decreasing sequence is another type of monotonic sequence where each subsequent term is less than or equal to the previous term. Formally, a sequence \( \{a_n\} \) is non-increasing if for all \( n \), the inequality \( a_{n+1} \leq a_n \) holds true.
In the case of a strictly decreasing sequence, every term is less than the term before it, satisfying \( a_{n+1} < a_n \).
  • Example: The sequence 5, 3, 1, -1,... is strictly decreasing since each term decreases by 2.
For a sequence to be deemed decreasing, it must consistently follow this non-increasing pattern throughout all of its terms, ensuring a steady downward trajectory.
Mathematical proof
Mathematical proof is a logical argument that establishes the truth of a given statement. It involves systematically demonstrating that certain conditions or assertions lead to a conclusion.
In the context of sequences, to prove whether a sequence is monotone, one can check whether it fulfills the criteria for being non-decreasing or non-increasing, as discussed in the respective sections on increasing and decreasing sequences.
A proof typically involves:
  • Assumptions: Identifying given conditions or facts that are considered true.
  • Logical Sequence: Using established rules and logical reasoning to connect the facts.
  • Conclusion: Arriving at a statement that definitively establishes whether the sequence is monotonic.
For instance, in examining the sequence condition \( a_{n+1} - a_n eq 0 \), the proof shows it doesn鈥檛 ensure monotonicity because it doesn鈥檛 maintain a consistent increase or decrease for all terms.

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Most popular questions from this chapter

Prove: (a) If \(f\) is an even function, then all odd powers of \(x\) in its Maclaurin series have coefficient \(0 .\) (b) If \(f\) is an odd function, then all even powers of \(x\) in its Maclaurin series have coefficient \(0 .\)

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