Problem 78 - Calculus Early Transcendentals

Question

Find the limit of the following sequences or state that the limit does not exist. $$\begin{aligned} &\left\\{\frac{1}{2}, \frac{4}{3}, \frac{9}{4}, \frac{16}{5}, \ldots\right\\}, \text { which is defined by } f(n)=\frac{n^{2}}{n+1}, \text { for }\\\ &n=1,2,3, \dots \end{aligned}$$

Step-by-Step Solution

Answer

Answer: The limit of the sequence is 1.

Step 1: Define the sequence function

The sequence is defined by the function $$f(n) = \frac{n^2}{n+1}$$ for positive integers n (1, 2, 3, ...). This means that the sequence of terms is $$\left\\{\frac{1^2}{1+1}, \frac{2^2}{2+1}, \frac{3^2}{3+1}, \frac{4^2}{4+1}, ...\right\\}$$

Step 2: Check the first few terms

To get a better understanding of the sequence, let's evaluate the first few terms using the function $$f(n) = \frac{n^2}{n+1}$$. For n = 1: $$f(1) = \frac{1^2}{1+1} = \frac{1}{2}$$\ For n = 2: $$f(2) = \frac{2^2}{2+1} = \frac{4}{3}$$\ For n = 3: $$f(3) = \frac{3^2}{3+1} = \frac{9}{4}$$\ For n = 4: $$f(4) = \frac{4^2}{4+1} = \frac{16}{5}$$

Step 3: Apply limit definition and finding the limit as n approaches infinity

We are trying to find the limit of this sequence as n approaches infinity: $$\lim_{n\to\infty} \frac{n^2}{n+1}$$. To find the limit, it can be helpful to divide the numerator and the denominator by the highest power of n, which in this case is n^2: $$\lim_{n\to\infty} \frac{n^2}{n+1} = \lim_{n\to\infty} \frac{n^2/n^2}{(n+1)/n^2}$$

Step 4: Simplify the expression

After dividing the numerator and the denominator by $$n^2$$, we can simplify the expression in the limit as follows: $$\lim_{n\to\infty} \frac{1}{\frac{1}{n}+\frac{1}{n^2}}$$ As n approaches infinity, the terms $$\frac{1}{n}$$ and $$\frac{1}{n^2}$$ go to 0, and the expression simplifies to: $$\lim_{n\to\infty} \frac{1}{0+0} = 1$$

Step 5: State the final result

The limit of the sequence defined by $$f(n) = \frac{n^2}{n+1}$$ as n approaches infinity is 1. Therefore, the limit exists, and it is equal to 1.

Key Concepts

Sequences

Sequences are ordered lists of numbers following a particular rule. In mathematics, sequences are vital for understanding patterns and behaviors of numbers. Take the example of the sequence defined by the function \( f(n) = \frac{n^2}{n+1} \). The numbers produced from this function form our sequence: \( \left\{ \frac{1}{2}, \frac{4}{3}, \frac{9}{4}, \frac{16}{5}, \ldots \right\} \). Here, each term is computed with positive integer values of \( n \), such as 1, 2, 3, and so on.

Understanding sequences involves:

• Evaluating initial terms to observe behavior, as was shown in the exercise with the values \( n = 1, 2, 3, 4 \).
• Determining patterns or regularity in the terms.

This sequence exhibits quadratic growth as \( n \) becomes large; each term seems to edge closer to a certain value, which points towards the concept of limits.

Infinity

Infinity in mathematics often represents an unbounded quantity that cannot be expressed as a real number. When considering the limits of sequences, we frequently use the concept of infinity to describe behavior as \( n \) grows larger without bound. In the sequence \( f(n) = \frac{n^2}{n+1} \), we calculate the limit as \( n \to \infty \).

Looking at infinity, consider:

• How terms behave as \( n \) increases. Here, both the numerator (\( n^2 \)) and denominator (\( n+1 \)) increase significantly, but at different rates.
• What aspects of infinity help simplify equations, such as assuming small terms \( \frac{1}{n}, \frac{1}{n^2} \) approach zero.

Infinity aids in finding that the sequence nears a specific limit, 1, even as its elements themselves continue to grow larger.

Calculus

Calculus is a branch of mathematics dealing with limits, derivatives, and integrals. It provides tools for finding the behavior of functions and sequences as inputs change, particularly as they increase or decrease without bounds. In our sequence example, the concept of limits from calculus is used to determine \( \lim_{n\to\infty} \frac{n^2}{n+1} \).

Relevant calculus steps for limits include:

• Rewriting expressions for more straightforward limit evaluation, for instance, dividing both the numerator and the denominator by the highest power of \( n \).
• Identifying terms that approach zero, simplifying the limit process, such as the terms \( \frac{1}{n} \) and \( \frac{1}{n^2} \) converging to zero as \( n \to \infty \).

By using these calculus principles, we deduce that the limit of the sequence is 1, signifying that the sequence stabilizes at this point as \( n \) becomes infinitely large.