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Chapter 2: Problem 1

Which of the following sequences are non-decreasing? non-increasing? bounded? (a) \(\frac{1}{n}\) (b) \(\frac{(-1)^{n}}{n^{2}}\) (c) \(n^{5}\) (d) \(\sin \left(\frac{n \pi}{7}\right)\) (e) \((-2)^{n}\) (f) \(\frac{n}{3^{n}}\)

Short Answer

Expert verified
(a) non-increasing and bounded, (b) bounded, (c) non-decreasing, (d) bounded, (e) neither, (f) non-increasing and bounded.

Step by step solution

01

Analyze sequence \(\frac{1}{n}\)

For this sequence, as n increases, the value of \(\frac{1}{n}\) decreases. It is a strictly decreasing sequence. It is bounded below by 0.
02

Analyze sequence \(\frac{(-1)^{n}}{n^{2}}\)

This sequence alternates between positive and negative values depending on the parity of n. The magnitude of the terms decreases as n increases. The sequence is bounded and non-increasing when considering the magnitude, but not monotonous overall.
03

Analyze sequence \(n^{5}\)

For this sequence, each term grows significantly larger as n increases. It is strictly increasing and unbounded.
04

Analyze sequence \(\sin \left(\frac{n \pi}{7}\right)\)

This sequence oscillates between -1 and 1, and does not have a monotonous nature. It is bounded by -1 and 1.
05

Analyze sequence \((-2)^{n}\)

This sequence changes sign with each term and grows in magnitude. It is neither bounded nor monotonous.
06

Analyze sequence \(\frac{n}{3^{n}}\)

As n increases, the term \(\frac{n}{3^{n}}\) decreases exponentially to 0, making it strictly decreasing. It is also bounded below by 0.

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Key Concepts

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

Non-Decreasing Sequences
In calculus, a sequence is called non-decreasing if each term is greater than or equal to the previous one. Formally, a sequence \((a_n)\) is non-decreasing if for all \(n\), \(a_{n+1} \geq a_n\). This means the sequence either stays the same or increases as \(n\) gets larger.
Examples include:
  • \(a_n = n\): Here, each term is greater than the last, making the sequence non-decreasing.
  • \(a_n = 2\): This constant sequence remains the same for all terms, hence it's non-decreasing.
In the given exercise, none of the sequences listed are non-decreasing. Let's look briefly at each one:
  • \(\frac{1}{n}\): This is strictly decreasing, not non-decreasing.
  • \(\frac{(-1)^n}{n^2}\): This alternates and decreases in magnitude.
  • \(n^5\): This is strictly increasing.
  • \(\sin \left( \frac{n \pi}{7} \right)\): This is oscillatory.
  • \((-2)^n\): This alternates in sign and increases in magnitude.
  • \(\frac{n}{3^n}\): This is strictly decreasing.
Non-Increasing Sequences
A sequence is non-increasing if each term is less than or equal to the previous one. Formally, a sequence \((a_n)\) is non-increasing if for all \(n\), \(a_{n+1} \leq a_n\). This implies the sequence either stays the same or decreases as \(n\) gets larger.
Examples include:
  • \(a_n = -n \): Each term is less than the last, making it non-increasing.
  • \(a_n = 5\): This constant sequence remains the same for all terms; it's non-increasing.
In the exercise, two sequences meet this criterion:
  • \(\frac{1}{n}\): This sequence is strictly decreasing, therefore non-increasing.
  • \(\frac{n}{3^n}\): This sequence decreases exponentially and is non-increasing.
Other sequences do not meet this definition:
  • \(\frac{(-1)^n}{n^2}\): Alternates in sign and decreases in magnitude but not simply non-increasing overall.
  • \(n^5\): This is strictly increasing.
  • \(\sin \left( \frac{n \pi}{7} \right)\): This is oscillatory.
  • \((-2)^n\): Alternates in sign and magnitude, not monotonous.
Bounded Sequences
A sequence is bounded if there exists a number that all terms of the sequence lie within. Specifically, a sequence \((a_n)\) is bounded if there exist numbers \(M\) and \(m\) such that \(m \leq a_n \leq M\) for all \(n\).
Simply put:
  • If you can find a box around the sequence, it's bounded.
In our exercise:
  • \(\frac{1}{n}\): Bounded below by 0.
  • \(\frac{(-1)^n}{n^2}\): Bounded since it alternates and decreases in magnitude.
  • \(n^5\): This is unbounded as terms grow infinitely large.
  • \(\sin \left( \frac{n \pi}{7} \right)\): Bounded by \([-1, 1]\).
  • \((-2)^n\): Unbounded as magnitude grows infinitely.
  • \(\frac{n}{3^n}\): Bounded below by 0 as it trends towards 0.
Monotonic Sequences
A sequence is considered monotonic if it is either entirely non-decreasing or non-increasing. In simpler terms, a monotonic sequence doesn鈥檛 change its direction; it keeps going up or down.
Examples include:
  • \(a_n = n\): This sequence is strictly increasing, hence monotonic.
  • \(a_n = -n\): This sequence is strictly decreasing, and also monotonic.
  • \(a_n = 5\): A constant sequence, remaining the same; it is both non-decreasing and non-increasing, thus monotonic.
Based on the exercise:
  • \(\frac{1}{n}\): Strictly decreasing; it's a monotonic decreasing sequence.
  • \(\frac{(-1)^n}{n^2}\): Not monotonic; alternates in sign.
  • \(n^5\): Strictly increasing, hence monotonic.
  • \(\sin \left( \frac{n \pi}{7} \right)\): Oscillatory, not monotonic.
  • \((-2)^n\): Not monotonic as terms change signs and magnitude.
  • \(\frac{n}{3^n}\): Strictly decreasing, hence monotonic.
Sequence Oscillation
Sequence oscillation refers to sequences that do not settle into a single pattern of growth or decay but instead move back and forth. These sequences change direction frequently and are not monotonic.
Examples include:
  • \(\sin(n)\): Oscillates between \( -1 \) and \( 1 \).
  • \((-1)^n\): Alternates between \(1\) and \(-1\).
From the exercise:
  • \(\sin \left( \frac{n \pi}{7} \right)\): Oscillates between \(-1\) and \(1\), demonstrating clear oscillatory behavior.
  • \((-2)^n\): Alternates in sign and magnitude, hence oscillatory.
  • Other sequences do not show oscillation as they follow a clear increasing or decreasing trend.

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Most popular questions from this chapter

Let \(\gamma_{n}=\left(\sum_{k=1}^{n} \frac{1}{k}\right)-\log _{e} n=\sum_{k=1}^{n} \frac{1}{k}-\int_{1}^{n} \frac{1}{t} d t\) (a) Show that \(\left(\gamma_{n}\right)\) is a decreasing sequence. Hint: Look at \(\gamma_{n}-\gamma_{n+1}\) (b) Show that \(0<\gamma_{n} \leq 1\) for all \(n\) (c) Observe that \(\gamma=\lim _{n} \gamma_{n}\) exists and is finite.

Let \(\left(s_{n}\right)\) be a bounded sequence in \(\mathbb{R}\). Let \(A\) be the set of \(a \in \mathbb{R}\) such that \(\left\\{n \in \mathbb{N}: s_{n}b\right\\}\) is finite. Prove that \(\sup A=\lim \inf s_{n}\) and \(\inf B=\lim \sup s_{n}\).

Let \(B\) be the set of all bounded sequences \(x=\left(x_{1}, x_{2}, \ldots\right),\) and define \(d(x, y)=\sup \left\\{\left|x_{j}-y_{j}\right|: j=1,2, \ldots\right\\}\) (a) Show that \(d\) is a metric for \(B\) (b) Does \(d^{*}(x, y)=\sum_{j=1}^{\infty}\left|x_{j}-y_{j}\right|\) define a metric for \(B ?\)

Let \(s_{1}=1\) and \(s_{n+1}=\left(\frac{n}{n+1}\right) s_{n}^{2}\) for \(n \geq 1\) (a) Find \(s_{2}, s_{3}\) and \(s_{4}\) (b) Show that \(\lim s_{n}\) exists. (c) Prove that \(\lim s_{n}=0\)

(a) Prove that if \(\lim s_{n}=+\infty\) and \(\lim \inf t_{n}>0,\) then \(\lim s_{n} t_{n}=\) \(+\infty\). (b) Prove that if \(\lim \sup s_{n}=+\infty\) and \(\lim \inf t_{n}>0,\) then \(\limsup s_{n} t_{n}=+\infty\). (c) Observe that Exercise 12.7 is the special case of (b) where \(t_{n}=k\) for all \(n \in \mathbb{N}\).
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