Problem 29 - Essential Calculus

Question

In Exercises \(29-32,\) use a graphing utility to graph the first 10 terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your inference analytically and, if the sequence converges, find its limit. \(a_{n}=\frac{n+1}{n}\)

Step-by-Step Solution

Answer

The sequence \(a_{n}=\frac{n+1}{n}\) converges and its limit is 1.

Step 1: Graphing the first 10 terms

The first 10 terms of the sequence \(a_{n}=\frac{n+1}{n}\) are calculated and plotted using a graphing utility. The x-axis represents the term number 'n' and the y-axis represents the sequence value \(a_{n}\). It's observed in the plot whether the sequence appears to converge to a certain value, which means that the points get closer and closer to a certain y-value, or diverges, which means that the points do not get closer to any specific y-value and keep moving in a certain direction or arbitrarily.

Step 2: Analytical Verification

After making a conjecture based on the graph, an analytical approach is employed to verify it. For the sequence \(a_{n}=\frac{n+1}{n}\) in particular, it can be simplified to \(1 + \frac{1}{n}\), as 'n' tends to infinity this approaches to \(1 + 0\). Hence, the analitycal limit is 1. This confirms whether the sequence does in fact converge to that value or not.

Step 3: Conclusion

Based on the graphical observation and analytical verification, the conclusion about whether the given sequence converges or diverges is stated. In case of sequence \(a_{n}=\frac{n+1}{n}\), if it's found out that the sequence converges, and to the limit 1, then this forms the final statement

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools to visually represent the behavior of sequences. They help in plotting the terms of a sequence on a graph, making it easier to infer whether the sequence converges or diverges. For the sequence given, \(a_{n} = \frac{n+1}{n}\), the graph is plotted with the term number \(n\) on the x-axis and the value of the sequence \(a_{n}\) on the y-axis.

• When plotting the first 10 terms of this sequence, each point corresponds to a specific term.
• The graph helps you visually inspect whether the plotted points seem to approach a specific y-value.
• If the points become closer to a certain value, the sequence converges to that value.
• If the points continue away without approaching a specific value, the sequence diverges.

Graphing utilities simplify the initial assessment of convergence or divergence, providing a preliminary visual estimate before any calculations are done.

Sequence Limit

The limit of a sequence is the value the sequence approaches as the term number \(n\) increases indefinitely. In simpler terms, it’s the number that the values of the sequence get closer to as \(n\) gets larger. For the sequence \(a_{n} = \frac{n+1}{n}\), we suspect if it converges, it should have a limit.

• By rewriting the expression, \(a_{n} = 1 + \frac{1}{n}\), it's seen in a form that makes it easier to find the limit.
• As \(n\) becomes very large, or approaches infinity, the term \(\frac{1}{n}\) becomes very small, approaching zero.
• This means that the term \(1 + \frac{1}{n}\) approaches \(1 + 0 = 1\).

Thus, the limit of the sequence is 1, suggesting the sequence converges to this value, as it doesn't drift off to infinity or oscillate erratically.

Analytical Verification

Analytical verification involves using mathematical tools and logic to confirm the behavior of a sequence that a graph suggests. For our sequence, we've assumed from the graph that it might converge, and by simplifying \(a_{n} = \frac{n+1}{n}\) to \(1 + \frac{1}{n}\), we have a clearer path for verification.

• By analyzing this simplified form, it is evident that as \(n\to\infty\), the term \(\frac{1}{n}\to 0\).
• Accordingly, \(a_{n}\) approaches \(1\), as the extraneous fraction diminishes.
• This confirms that graphically observed convergence is correct, and mathematically, the sequence has a limit of 1.

Analytical verification solidifies the graphical inference, ensuring that the behavior isn’t a misconception due to graphical inaccuracies or plot errors. It gives a definitive mathematical backbone to our initial observations.