Problem 91 - 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. \(\left\\{2, \frac{3}{4}, \frac{4}{9}, \frac{5}{16}, \ldots\right\\},\) which is defined by \(f(n)=\frac{n+1}{n^{2}},\) for \(n=1,2,3, \ldots\)

Step-by-Step Solution

Answer

Answer: The limit of the sequence is \(0\).

Step 1: Write down the function

We are given the function defining the sequence: $$f(n)=\frac{n+1}{n^2}$$ for \(n=1,2,3,\ldots\).

Step 2: Find the limit as n approaches infinity

Now, we need to find the limit of the function \(f(n)\) as \(n\) goes to infinity. To do this, we calculate the limit: $$\lim_{n\to\infty} \frac{n+1}{n^2}$$.

Step 3: Divide the numerator and denominator by the highest power of n in the denominator

Since both the numerator and denominator are polynomial functions, we can divide both by the highest power of \(n\) in the denominator, which in this case is \(n^2\). This gives us: $$\lim_{n\to\infty} \frac{\frac{n+1}{n^2}}{\frac{n^2}{n^2}}$$ Simplifying the expression, we get: $$\lim_{n\to\infty} \frac{\frac{1}{n}+\frac{1}{n^2}}{1}$$

Step 4: Find the limit of the simplified expression

Now, we can split the limit of the sum as the sum of the limits: $$\lim_{n\to\infty} \frac{1}{n} + \lim_{n\to\infty} \frac{1}{n^2}$$ As \(n\) approaches infinity, both terms in the expression go to zero: $$\lim_{n\to\infty} \frac{1}{n} = 0\quad\text{and}\quad\lim_{n\to\infty} \frac{1}{n^2} = 0$$ Thus, the limit of the sequence is: $$\lim_{n\to\infty} \frac{n+1}{n^2} = 0$$ So, the limit of the sequence \(\left\\{2, \frac{3}{4}, \frac{4}{9}, \frac{5}{16}, \ldots\right\\}\) is \(0\).

Key Concepts

Convergence of Sequences

The concept of convergence in sequences is fundamental to understanding limits in calculus. When we say that a sequence converges, we mean that as the sequence progresses, its terms approach a specific value, known as the limit. In the provided example, \(f(n) = \frac{n+1}{n^2}\), we're looking at how the sequence behaves as \(n\) gets larger and larger. We see that for very large values of \(n\), the \(\frac{n+1}{n^2}\) expression gets closer and closer to zero, indicating convergence to zero. In general, a sequence \(\{a_n\}\) is said to converge to limit \(L\) if for every positive number \(\epsilon\), however small, there exists a corresponding integer \(N\) such that \(\left| a_n - L \right| < \epsilon\) for all \(n > N\). Convergence is an assurance that a sequence becomes stable as it moves towards infinity.

Limit Laws for Sequences

Limit laws for sequences are the rules that allow us to manipulate and compute the limits of sequences methodically. These laws are invaluable tools for solving problems involving limits. The problem presented makes use of these laws to simplify and evaluate the limit. One such law is the ability to break apart a limit involving a sum, difference, or product into separate limits for each part of the sum, difference, or product. In our example, we applied it by separating \(\lim_{n\to\infty} \frac{1}{n} + \lim_{n\to\infty} \frac{1}{n^2}\). Another common limit law involves dividing terms by the highest power of \(n\) in the denominator to transform the terms into ones that are recognizable in their behavior at infinity — in this case transforming \(\frac{n+1}{n^2}\) into \(\frac{1}{n} + \frac{1}{n^2}\). Limit laws streamline the process of finding limits and make solving complex sequences manageable.

Limits at Infinity

Limits at infinity refer to the behavior of functions as the input grows without bound. Considering the sequence-based function \(f(n)\), we are interested in what happens to the function's output as \(n\) becomes larger — essentially as \(n \to \infty\). The limit describes the value that the function's output is approaching. For polynomial functions such as \(n+1\) or \(n^2\), when we divide by the highest power of \(n\) in each term as we did in the example solution, we facilitate the process of finding what the sequence approaches. It’s a common strategy used for sequences defined by ratios of polynomials. If as \(n\) approaches infinity, the function's value approaches some finite number, we say that the limit at infinity exists, like in our case, where the sequence converges to zero.

Polynomial Functions

Polynomial functions play a significant role in the study of sequences and their limits. A polynomial function is composed of terms which are multiples of powers of a variable. In the context of sequences, we might look at a function like \(f(n)\), which is used to generate the terms of the sequence. With polynomials, when computing limits at infinity, higher powers of \(n\) dominate the behavior of the function, which is why we focus on the highest power to simplify the limit evaluation. For instance, in our previous example, the presence of \(n^2\) in the denominator is pivotal. As \(n\) becomes very large, any term that isn’t involving the highest power becomes negligible. Therein lies the beauty of polynomial functions when dealing with limits: their predictable structure makes determining the behavior of a sequence as \(n \to \infty\) much simpler.