Problem 76 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: The limit of the sequence is 0.

Step 1: Evaluate the General Term at Some Larger n Values

As a first step to find the limit of the sequence, we can plug-in some larger n values into the general term formula, \(f(n) = \frac{4}{n}\), to see if the terms are approaching any value or not.

Step 2: Look for a Pattern

From the given sequence as well as the values calculated in step 1, we might notice that as \(n\) increases, the terms of the sequence seem to be getting smaller and closer to 0. This observation suggests a probable limit, and we can now attempt to find the limit.

Step 3: Calculate the Limit Using the General Term Formula

We can use the formula \(f(n) = \frac{4}{n}\) to find the limit as \(n\) goes to infinity. The limit is: $$\lim_{n \to \infty} \frac{4}{n}$$

Step 4: Simplify the Limit Expression

Since 4 is a constant term and the denominator (n) is approaching infinity, the whole expression will approach zero: $$\lim_{n \to \infty} \frac{4}{n} = \frac{4}{\infty} = 0$$

Step 5: State the Final Answer

From the above analysis, we found that the limit of the given sequence exists, and it is equal to 0 as \(n\) goes to infinity. Therefore, the limit of the sequence \(\{4, 2, \frac{4}{3}, 1, \frac{4}{5}, \frac{2}{3}, \ldots\}\) is 0.

Key Concepts

Limit of a Sequence

In mathematics, the limit of a sequence is a crucial concept that helps us understand the behavior of sequences as they progress. Consider a sequence \(a_n = \frac{4}{n}\). When we talk about finding the limit, we're interested in what value \(a_n\) approaches as \(n\) becomes very large.

In our example, as \(n\) increases, each term in the sequence \(\left\{4, 2, \frac{4}{3}, 1, \frac{4}{5}, \frac{2}{3}, \ldots\right\}\) is calculated by dividing 4 by the term number \(n\). A pattern emerges: these terms become smaller and tend towards zero.

To find the limit mathematically, we use the expression \(\lim_{n \to \infty} \frac{4}{n}\). This limit equals 0 because as \(n\) grows, \(\frac{4}{n}\) gets infinitesimally small. Thus, through this simple evaluation, we see that the limit of the sequence is 0.

Convergent Sequences

A sequence is said to be convergent if its terms approach a specific value as \(n\) becomes very large. If we revisit our example sequence \(a_n = \frac{4}{n}\), we see that as \(n\) increases, the terms converge to the number 0.

Convergence implies that regardless of how far along the sequence you are, the difference between the term and the limit becomes arbitrarily small. It's like aiming at a target and getting closer with each step, but never missing.

Recognizing convergent sequences is vital in calculus because they lead to stable results. Non-convergent sequences either oscillate or drift and don't settle to a single limit.

Calculus

Calculus is a branch of mathematics focusing on limits, functions, derivatives, integrals, and infinite series. Understanding sequences is foundational in calculus, especially when calculating limits and analyzing convergent behavior.

When dealing with sequences in calculus, finding the limit is akin to discovering the function's behavior as the variable approaches infinity or some other point of interest. This skill is indispensable in differential and integral calculus, where we often encrypt complex real-world problems into manageable mathematical models.

In our task, limits help identify how a sequence behaves over time. They also prepare us for more complex topics such as series summation and integral approximation. Thus, sequences offer us a window into the perennial frontier of calculus, informing both theoretical and practical applications.