Problem 89 - Calculus: Early Transcendentals

Question

Looking ahead to sequences A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence \(\\{2,4,6,8, \ldots\\}\) is specified by the function \(f(n)=2 n,\) where \(n=1,2,3, \ldots . .\) The limit of such a sequence is \(\lim _{n \rightarrow \infty} f(n),\) provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist. \(\left\\{0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots\right\\},\) which is defined by \(f(n)=\frac{n-1}{n},\) for \(n=1,2,3, \ldots\)

Step-by-Step Solution

Answer

Answer: The limit of the sequence as \(n\) approaches infinity is 1.

Step 1: Identify the function and variables

We are given the sequence \(\left\{0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots\right\}\) defined by the function \(f(n)=\frac{n-1}{n}\) for \(n=1,2,3, \ldots\)

Step 2: Find the limit of the function at infinity

We need to find the limit of the function as \(n\) approaches infinity: \(\lim_{n \to \infty} \frac{n-1}{n}\). We can do this by applying the limit laws for limits at infinity and simplifying the function.

Step 3: Simplify the function

We have: $$\lim_{n \to \infty} \frac{n-1}{n}$$ Divide both numerator and denominator by \(n\) to get: $$\lim_{n \to \infty} \frac{\frac{n}{n}-\frac{1}{n}}{1} = \lim_{n \to \infty} (\frac{n}{n}-\frac{1}{n})$$

Step 4: Apply the limit laws for limits at infinity

Now we can apply the limit laws for limits at infinity: $$\lim_{n \to \infty} (\frac{n}{n}-\frac{1}{n}) = \lim_{n \to \infty} \frac{n}{n} - \lim_{n \to \infty} \frac{1}{n}$$

Step 5: Evaluate the limits

Evaluate the limits: - \(\lim_{n \to \infty} \frac{n}{n} = 1\). It is because as we increase \(n\), the ratio between the numerator and denominator remains one. - \(\lim_{n \to \infty} \frac{1}{n} = 0\). It is because as we increase \(n\), the value of \(\frac{1}{n}\) goes closer and closer to 0.

Step 6: Calculate the final limit

So, we have: $$\lim_{n \to \infty} (\frac{n}{n}-\frac{1}{n}) = 1 - 0$$ Therefore, the limit of the sequence as \(n\) approaches infinity is \(\boxed{1}\).

Key Concepts

Convergence of Sequences

Understanding the behavior of sequences as they extend towards infinity is pivotal in calculus. When discussing the convergence of sequences, we are essentially asking whether, as the terms of the sequence get larger and larger, they approach a specific value. This value is known as the limit of the sequence.

For example, when dealing with the sequence \(0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots\), we notice that as the value of \(n\) increases, the terms of the sequence grow closer to 1. In mathematical terms, if a sequence \(a_n\) approaches a number \(L\) as \(n\) approaches infinity, we say that \(L\) is the limit of the sequence, and that the sequence converges to \(L\). If no such \(L\) exists, the sequence is said to diverge.

To confirm this behavior, we can examine the sequence algebraically by using strategies such as simplifying the terms and utilizing limit laws. The sequence in question simplifies to \(\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)\), which, upon evaluation, converges to 1.

Infinite Series

While a sequence is a list of numbers, an infinite series represents the sum of the elements of a sequence. It is generally expressed as the summation of terms starting from an initial index and continuing indefinitely. For the purpose of this discussion and relating to the given sequence, an infinite series isn't directly the subject, but it's important to understand that concepts regarding convergence apply to both sequences and series.

Convergence in the context of series would mean that the infinite sum approaches a finite value. Notice that this is similar in spirit to the convergence of sequences, where the terms approach a finite limit. The ability to determine whether an infinite series converges or diverges is crucial, especially when dealing with power series or when calculating the sum of an infinite geometric series. However, for the present sequence, our focus is the value each term of the sequence approaches rather than the sum of the terms.

Limit Laws for Infinity

To tackle limits involving infinity, mathematicians have developed a series of rules or limit laws for infinity. These laws offer a structured approach to determine the limit of a function as the variable approaches infinity. Important to the calculation of our sequence's limit is the law that the limit of a constant divided by infinity is zero.

In the context of the given sequence, these laws help us break down the limit into more manageable parts. To use these laws effectively, we often transform the sequence's terms into a form that makes applying the laws straightforward. For the sequence \(0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots\), we divided the numerator and the denominator by \(n\) and applied the laws to find that as \(n\) approaches infinity, the limit is 1. This application of limit laws is a powerful tool in determining the behavior of sequences and functions at infinity.