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Chapter 2: Problem 5

Show that a strictly increasing sequence has no peak indices.

Short Answer

Expert verified
In a strictly increasing sequence, each subsequent term is greater than the previous one, contradicting the requirement for a peak index, thus proving there can be no peak indices in such a sequence.

Step by step solution

01

Define Strictly Increasing Sequence

A strictly increasing sequence is one in which each term is larger than the previous term. In mathematical terms, a sequence \(a_n\) is strictly increasing if for all \(n \) in the natural numbers, \(a_{n+1} > a_n\).
02

Define Peak Index

A peak index in a sequence is an index \(i\) such that the terms on both sides of \(a_i\) are less than or equal to \(a_i\). Formally, a peak index \(i\) satisfies \(a_{i-1} \leq a_i\) and \(a_{i+1} \leq a_i\), if \(i-1\) and \(i+1\) are valid indices within the sequence.
03

Explain Absence of Peak Indices in a Strictly Increasing Sequence

In a strictly increasing sequence, for every index \(i\), the term that follows is strictly greater, \(a_{i+1} > a_i\), which contradicts the definition of a peak index, as a peak index requires \(a_{i+1} \leq a_i\). Therefore, a strictly increasing sequence cannot have a peak index because it is never true that a term will be less than or equal to the previous term.

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Key Concepts

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

Peak Index
The term 'peak index' might evoke images of mountaintop panoramas, but in the landscape of mathematics, it has a very precise definition. A peak index refers to a specific position in a sequence of numbers where the value at that index is not smaller than the values of its immediate neighbors. Imagine standing at a peak: to either side, the paths lead downhill - it's the same in a sequence. To be called a peak, the term must satisfy the condition that both neighboring terms are less than or equal to the term at the peak index.

More formally, in a sequence labeled as \( a_i \) with indices from the set of natural numbers, an index \( i \) is considered a peak if and only if \( a_{i-1} \leq a_i \) and \( a_{i+1} \leq a_i \) when \(i-1\) and \(i+1\) are valid indices within the sequence. It's important to realize that not every sequence will have a peak index. As we delve into strictly increasing sequences, you'll discover why these kinds of series don't allow for a scenic 'peak' at any point.
Sequence in Mathematics
Diving into the depths of mathematics reveals the concept of a 'sequence,' a list of numbers arranged in a specific order. Sequences can be finite, having a clear beginning and end, or infinite, stretching out to infinity. Each number in the sequence is called a 'term.' We denote the terms of the sequence with a subscript as \( a_n \) where \( n \) represents the index of the term, which corresponds to its position in the sequence.

Mathematicians have identified many types of sequences, among which are arithmetic sequences where terms increase by a constant difference, geometric sequences that multiply by a constant factor, and many others. Sequences can tell stories of growth, decay, or even random fluctuations. The strictly increasing sequence is a sequence where each term is greater than the term that came before it - no repeats and no backtracking. It symbolizes unceasing progress, like climbing a ladder where each step takes you higher than the last.
Natural Numbers
Natural numbers are the building blocks of many mathematical concepts. They are as intuitive as counting on your fingers starting from 1, 2, 3, and so on, extending to infinity. Unlike whole numbers, natural numbers do not include zero. They are often used to count items, label positions, or indicate the size of a set in an uncomplicated way because they are free from fractions and decimals.

Natural numbers come in handy when working with sequences. Each term in a sequence is often linked to a natural number as its index, indicating the term's position. The properties of natural numbers, such as their order and additive nature, provide a solid foundation for various operations in sequence analysis. The clear incremental nature of natural numbers aligns perfectly with the concept of a strictly increasing sequence, where each subsequent term follows a clear, step-by-step progression.

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Most popular questions from this chapter

For a pair of positive numbers \(\alpha\) and \(\beta,\) the number \(\sqrt{\alpha \beta}\) is called the geometric mean of \(\alpha\) and \(\beta,\) and the number \((\alpha+\beta) / 2\) is called the arithmetic mean of \(\alpha\) and \(\beta\). By observing that \((\sqrt{\alpha}-\sqrt{\beta})^{2} \geq 0,\) show that \((\alpha+\beta) / 2 \geq \sqrt{\alpha \beta}\).

Let \(\left\\{a_{n}\right\\}\) be a sequence of real numbers. Suppose that for each positive number \(c\) there is an index \(N\) such that $$a_{n}>c$$ for all indices \(n \geq N\) When this is so, the sequence \(\left\\{a_{n}\right\\}\) is said to converge to infinity, and we write $$\lim _{n \rightarrow \infty} a_{n}=\infty$$ Prove the following: a. \(\lim _{n \rightarrow \infty}\left[n^{3}-4 n^{2}-100 n\right]=\infty\) $$\text { b. } \lim _{n \rightarrow \infty}\left[\sqrt{n}-\frac{1}{n^{2}}+4\right]=\infty$$

Let \(a\) and \(b\) be numbers with \(a

Let \(S\) be the set of rational numbers in the interval [0,2] . a. Using the definition of sequential compactness, show that \(S\) is not sequentially compact. b. Using the definition of compactness, show that \(S\) is not compact. c. Using the definition of closedness, show that \(S\) is not closed.

Prove that the sequence \(\left\\{c_{n}\right\\}\) converges to \(c\) if and only if the sequence \(\left\\{c_{n}-c\right\\}\) converges to 0 .
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