Problem 52 - Precalculus with Limits

Question

In Exercises 49-52, find the limit of the sequence. Then verify the limit numerically by using a graphing utility to complete the table. $$ a_n = \dfrac{n(n+1)}{n^2} - \dfrac{1}{n^4} \left[\dfrac{n(n+1)}{2} \right]^2 $$

Step-by-Step Solution

Answer

The limit of the sequence as 'n' approaches infinity is 1.

Step 1: Simplify the sequence

First, we simplify the sequence \( a_n = \dfrac{n(n+1)}{n^2} - \dfrac{1}{n^4} \cdot \left[\dfrac{n(n+1)}{2} \right]^2 \) as much as possible. The first term, \( \dfrac{n(n+1)}{n^2} \), reduces to \( 1 + \dfrac{1}{n} \). And the second term, \( \dfrac{1}{n^4} \cdot \left[\dfrac{n(n+1)}{2} \right]^2 \) simplifies to \( \dfrac{1}{4n^2} \cdot (1 + \dfrac{1}{n})^2 \). So the sequence becomes \( a_n = 1 + \dfrac{1}{n} - \dfrac{1}{4n^2} \cdot (1 + \dfrac{1}{n})^2 \).

Step 2: Calculate the limit

Now we can calculate the limit as n approaches infinity. When n approaches infinity, \( \dfrac{1}{n} \) goes to 0. Hence, our sequence becomes \( a_n = 1 + 0 - 0 = 1 \). Hence the limit of the sequence is 1.

Step 3: Verification using a graphing utility

To validate these findings numerically, using a graphing utility software such as Desmos or GeoGebra, plot the simplified sequence in terms of n. As the value of n increases, it can be observed that the function values converge to 1 thus implying that the limit of the sequence as n approaches infinity is indeed 1.

Key Concepts

Sequences

Sequences are essentially ordered lists of numbers. In mathematics, they are used to observe patterns and analyze the behavior of functions over time. Each number in the sequence is called a term. We often denote sequences with notation like \( a_n \), where \( n \) is an index that usually represents natural numbers (1, 2, 3, etc.).
Understanding a sequence means looking at how each term is defined and how the terms relate to each other. Sequences may become increasingly complex, as in our exercise.
By observing the expression for \( a_n \) in its simplified form, we can see that each part of the equation contributes to the behavior of \( a_n \) as \( n \) becomes large. This process of simplification is crucial to finding limits of sequences, which we will discuss in the next sections.

Graphing Utility

A graphing utility is an essential tool in modern mathematics education. Tools like Desmos or GeoGebra allow users to visually analyze mathematical functions and sequences. When dealing with sequences, graphing utilities provide a way to observe how terms behave as the index \( n \) changes.
In our sequence exercise, plotting \( a_n = 1 + \frac{1}{n} - \frac{1}{4n^2} \cdot (1 + \frac{1}{n})^2 \) using a graphing utility helps verify the calculated limit. When you graph this sequence, it visually approaches the value 1 as \( n \) increases, confirming the analytic solution.
Advantages of using graphing utilities include:

• Immediate visual confirmation of numerical calculations
• The ability to test different sequences quickly
• An interactive way to understand the convergence of sequences

By providing a visual representation, graphing utilities facilitate better understanding and provide tangible evidence to support mathematical findings.

Infinite Limits

Infinite limits involve understanding the behavior of sequences as they extend towards infinity. It's like zooming out on a number line and watching what happens as you move along to the very far right.
When we talk about the limit of a sequence as \( n \to \infty \), we are interested in what value, if any, the terms of the sequence approach. For example, with our given sequence \( a_n = 1 + \frac{1}{n} - \frac{1}{4n^2}\cdot (1+\frac{1}{n})^2 \), as \( n \) becomes very large, the terms like \( \frac{1}{n} \) and \( \frac{1}{4n^2} \cdot (1 + \frac{1}{n})^2 \) diminish or become negligible, leaving \( a_n \) to approach 1.
This concept is crucial in many areas of calculus and real analysis, as it helps to describe the behavior of functions and sequences at the limits of their domains. Understanding infinite limits also establishes a foundation for further study in series, integrals, and differentiated equations.