Problem 45 - Single Variable Calculus

Question

(a) graph the sequence \(\left\\{a_{n}\right\\}\) with a graphing utility, (b) use your graph to guess at the convergence or divergence of the sequence, and (c) use the properties of limits to verify your guess and to find the limit of the sequence if it converges. $$ a_{n}=\frac{n !}{n^{n}} $$

Step-by-Step Solution

Answer

In summary, we graphed the sequence \(a_n = \frac{n!}{n^n}\) and observed it approaching a finite value, indicating convergence. We then identified the ratio of consecutive terms as \(b_n = \frac{n^n}{(n+1)^n}\) and found the limit of this ratio as n approaches infinity, which is \(e^{-1}\). Since the ratio's limit is between 0 and 1, we confirmed that the sequence converges according to the ratio test.

Step 1: Graph the sequence

To begin, use a graphing utility to plot the points of the sequence \(a_n\). While different graphing tools may have different specific steps to follow, the general process includes entering the given function \(a_n = \frac{n!}{n^n}\) and ensuring that the input variable is "n." This will generate a graph of the sequence.

Step 2: Observe the graph for convergence/divergence

Now that we have the sequence's graph, we need to examine it for convergence or divergence. Observe the graph to see if it approaches a limiting value as n increases, which would indicate convergence. If not, the sequence diverges. In this case, the graph clearly shows that it is approaching a finite value. Hence, this is an indication of the convergence of the sequence.

Step 3: Determine the limit of the sequence

We can formally confirm our guess in Step 2 using the properties of limits and finding the limit as \(n\) approaches infinity. To do this, consider the ratio of consecutive terms in the sequence \(b_n = \frac{a_{n+1}}{a_n}\). We need to find the limit as n goes to infinity \(L = \lim_{n\to\infty} b_n\). First, we find the closed-form expression for \(b_n\): \(b_n = \frac{a_{n+1}}{a_n} = \frac{\frac{(n+1)!}{(n+1)^{n+1}}}{\frac{n!}{n^n}} = \frac{(n+1)!n^n}{n!(n+1)^{n+1}}\) Now, simplify the expression: \(b_n = \frac{(n+1)n^n}{(n+1)^{n+1}}\) \(b_n = \frac{n^n}{(n+1)^n}\) Next, we need to find the limit of \(b_n\) as n approaches infinity: \(\lim_{n\to\infty} \frac{n^n}{(n+1)^n} = \lim_{n\to\infty} \left(\frac{n}{n+1}\right)^n\) Let's use the change of variable \(k = \frac{1}{1 + 1/n}\), then: \(\lim_{n\to\infty} \left(\frac{n}{n+1}\right)^n = \lim_{k\to1} \left(1 - \frac{1}{\frac{1}{1-k}}\right)^{\frac{1}{1-k}}\) Using properties of limits, we evaluate the limit as follows: \[\lim_{k\to1} \left(1 - \frac{1}{\frac{1}{1-k}}\right)^{\frac{1}{1-k}} = \lim_{k\to1} (1-k)^{\frac{1}{1-k}} = e^{-1}\] Since \(0 < e^{-1} < 1\), by the ratio test, the sequence converges.

Step 4: Conclusion

We have verified that the graph was indeed suggesting convergence. Through the properties of limits, it has been confirmed that the sequence \(a_n = \frac{n!}{n^n}\) converges.

Key Concepts

Convergence and Divergence

Convergence and divergence are fundamental concepts when analyzing sequences and series. A sequence is said to converge if it approaches a specific value as the element number, often denoted by \( n \), increases indefinitely. This "approaching" value is known as the limit of the sequence. Conversely, a sequence diverges if it does not settle into a single, finite value as \( n \) becomes very large.
When examining a sequence for convergence, one might first look at a graph, as graphical insights can quickly hint at whether a sequence is converging to a finite number or diverging. For example, a sequence whose graph levels off horizontal as \( n \) increases is likely converging. However, a graphical approach alone cannot provide full certainty, which leads us to more rigorous mathematical checks such as calculating limits.
In the problem at hand, analyzing the behavior of the sequence \( a_{n} = \frac{n!}{n^n} \), it initially looks like the sequence converges. To confirm this suspicion, further analysis through formal mathematical tools, such as limit properties, is applied.

Limits

Limits play a crucial role in determining the behavior of sequences as \( n \) approaches infinity. Mathematically, a limit helps specify the value that a sequence gets close to, given particular conditions. It is defined as the value that a function (or sequence) approaches as the input (or index in sequences) grows without bound.
The limit of a sequence \( a_n \) as \( n \rightarrow \infty \) can be understood by examining the ratio of consecutive terms. In our problem, we explored \( b_n = \frac{a_{n+1}}{a_n} \) to understand how \( a_n \) behaves at large values of \( n \). By simplifying and taking the limit, it helped confirm if \( a_n \) actually settles to a finite value, thus converging.
For the given sequence \( a_{n} = \frac{n!}{n^n} \), we simplified and found the limit \( L \), which should be really small (in this case, \( e^{-1} \)). This result supported the fact that the original sequence does converge. Evaluating limits like these is crucial for confirming graphical insights about sequences.

Graphing Utilities

Graphing utilities are invaluable tools for visualizing mathematical concepts such as sequences and functions. They assist students and educators alike by providing a visual representation of a problem, which can often make understanding more intuitive.
To graph a sequence like \( a_{n} = \frac{n!}{n^n} \), a graphing utility typically requires inputting the sequence's formula and setting the variable to \( n \). This generates a sequence of points where you can observe patterns, convergence, or divergence. The visual aid of seeing how the points behave as \( n \) increases can offer hints about the sequence's behavior at infinity.
However, while graphing helps provide preliminary insights, it is not a replacement for mathematical rigor. It's always best to use graphs in conjunction with other methods, like computing limits, to achieve a deep and accurate understanding. Graphing utilities make the abstract more tangible and serve as a bridge between numerical data and theoretical mathematics.