Question

Find a real number \(c\) and an \(N_{0} \in \mathbb{N}\) such that \(n^{2}+3 n

Short answer

Choose \( c = 0.1 \) and \( N_0 = 11 \).

Step-by-step solution

Understand the Inequality

We need to find a real number \( c \) and a natural number \( N_0 \) such that the inequality \( n^2 + 3n < c n^3 \) holds for all natural numbers \( n \geq N_0 \).

Rewrite the Inequality

Divide both sides of the inequality \( n^2 + 3n < c n^3 \) by \( n^3 \) to simplify it to \( \frac{n^2}{n^3} + \frac{3n}{n^3} < c \), which simplifies further to \( \frac{1}{n} + \frac{3}{n^2} < c \).

Analyze the Simplified Inequality as n Approaches Infinity

As \( n \to \infty \), the terms \( \frac{1}{n} \) and \( \frac{3}{n^2} \) both approach 0. Therefore, the inequality \( \frac{1}{n} + \frac{3}{n^2} < c \) simplifies to \( 0 < c \).

Choose an Appropriate Value for c

Since we need \( c \) to be greater than 0 and work for large \( n \), choose \( c = 0.1 \) (or any small positive number) to ensure the inequality holds.

Determine N_{0} for the Chosen c

We need to find \( N_0 \) such that for all \( n \geq N_0 \), \( \frac{1}{n} + \frac{3}{n^2} < 0.1 \), or \( \frac{1}{n} < 0.1 - \frac{3}{n^2} \). As \( n \) increases, the right side approaches 0.1, making it possible to find an \( N_0 \).

Calculate N_{0}

For a practical \( N_0 \), let \( \frac{1}{n} < 0.1 \), giving \( n > 10 \). Also, verify with the term \( \frac{3}{n^2} \), which becomes negligible beyond \( n \). Choose \( N_0 = 11 \) to start satisfying the condition \( \frac{1}{11} + \frac{3}{11^2} < 0.1 \).

Key concepts

Real Numbers

Real numbers are a broad category of numbers that include all possible quantities along the number line. They consist of both rational numbers (like 3, 1/2, or -7) and irrational numbers (such as √2, π, or e). Real numbers can be thought of as an endless line where each point represents a unique number. This line encompasses all imaginable values, including natural numbers, integers, fractions, and numbers that cannot be expressed as simple fractions.
Understanding real numbers is crucial in mathematics as they allow for a comprehensive description of our world and are used extensively in calculations, equations, and inequalities. In the exercise provided, we search for a real number, denoted as 'c', which establishes a condition in an inequality involving 'n'.
Real numbers are an essential part of mathematical theories, including algebra, calculus, and beyond, where they help in understanding more complex structures.

Natural Numbers

Natural numbers are the set of positive integers starting from 1 and going upwards: 1, 2, 3, 4, and so forth. These numbers are the most basic and are often the first type of numbers people learn. They are used for counting objects and order things sequentially.
In mathematical contexts, the notation \( \mathbb{N} \) represents natural numbers. Sometimes, depending on the context, zero is included, but typically, natural numbers begin at 1. In the given exercise, we look for a natural number, \( N_{0} \), to restrict our inequality to work for values that are greater or equal to this specified number.
The concept of natural numbers helps build a foundation for more complex mathematical operations. They serve as the building blocks for whole numbers, integers, and are used widely in number theory and arithmetic.

Asymptotic Analysis

Asymptotic analysis is a key mathematical concept used to describe the behavior of functions as they tend towards infinity or some significant point. It often involves understanding how one function compares to another as a variable approaches a limit, often infinity.
In the context of the given exercise, asymptotic analysis helps us find the point where the inequality becomes true for all sufficiently large values of 'n'.

• We simplify the inequality to focus on terms that diminish as 'n' grows. This is why \( \frac{1}{n}\ \) and \( \frac{3}{n^2}\ \) become inconsequential as they approach zero.
• The goal is to choose an adequate value for 'c' that satisfies the inequality as 'n' increases, illustrating an instance of asymptotic bounds.

Through this analysis, we determine that the inequality holds true as 'n' surpasses a certain threshold (\( N_{0} = 11 \)). Thus, asymptotic analysis is pivotal in such proofs and helps in anticipating the performance and limits of mathematical functions.