Problem 84 - A Graphical Approach to Precalculus with Limits

Question

Use a graphing calculator to graph the first 10 terms of each sequence. Make a conjecture as to whether the sequence comerges or diverges. If you think it converges, make a conjecture about the mumber to which it converges. $$a_{n}=5-\frac{1}{n}$$

Step-by-Step Solution

Answer

The sequence converges to 5.

Step 1: Identify the terms of the sequence

The sequence is given by the formula \(a_n = 5 - \frac{1}{n}\). To find the first 10 terms, we plug in the integer values of \(n\) from 1 to 10 into the formula. This will give us the terms: \(a_1 = 5 - 1\), \(a_2 = 5 - 0.5\), \(a_3 = 5 - 0.333...\), and so on, up to \(a_{10} = 5 - 0.1\).

Step 2: Calculate the first 10 terms

Let's calculate the values for each \(n\): - \(n=1\), \(a_1 = 5 - 1 = 4\)- \(n=2\), \(a_2 = 5 - 0.5 = 4.5\)- \(n=3\), \(a_3 = 5 - 0.333 = 4.667\)- \(n=4\), \(a_4 = 5 - 0.25 = 4.75\)- \(n=5\), \(a_5 = 5 - 0.2 = 4.8\)- \(n=6\), \(a_6 = 5 - 0.167 = 4.833\)- \(n=7\), \(a_7 = 5 - 0.143 = 4.857\)- \(n=8\), \(a_8 = 5 - 0.125 = 4.875\)- \(n=9\), \(a_9 = 5 - 0.111 = 4.889\)- \(n=10\), \(a_{10} = 5 - 0.1 = 4.9\).

Step 3: Plot the terms on a graph

Take the calculated terms and plot them on a graph with \(n\) on the x-axis and \(a_n\) on the y-axis. The points to plot are (1, 4), (2, 4.5), (3, 4.667), (4, 4.75), (5, 4.8), (6, 4.833), (7, 4.857), (8, 4.875), (9, 4.889), (10, 4.9).

Step 4: Analyze the plotted graph

Observing the plotted points, notice how as \(n\) increases the value of \(a_n\) gets closer to 5. This suggests a trend where the terms of the sequence get closer to approaching a specific value.

Step 5: Make a conjecture about convergence

The terms of the sequence seem to be getting closer to 5 as \(n\) increases. Thus, we can conjecture that the sequence converges to 5.

Key Concepts

Graphing Calculators

Graphing calculators are powerful tools when it comes to visualizing mathematical sequences and understanding their behavior. By entering a sequence formula like \( a_n = 5 - \frac{1}{n} \), we can quickly calculate and plot numerous points of the sequence. This visualization helps in tracking the progression and identifying patterns among the terms. To use a graphing calculator effectively, follow these steps:

• Enter the sequence formula accurately.
• Specify the range of \( n \) values (in this exercise, \( n \) from 1 to 10).
• Graph the sequence to receive a visual representation of how the terms evolve as \( n \) increases.

Once the graph is plotted, observe how the terms position themselves on the grid. Notice especially if the values of \( a_n \) appear to approach a specific line or trend as \( n \) grows. This graphical representation can make analyzing the sequence much clearer than reviewing numerical calculations alone. The initial graph provides the first visual clue about the sequence's behavior, such as convergence.

Sequence Analysis

Sequence analysis involves examining the terms generated by a sequence formula to understand its progression. By following the formula \( a_n = 5 - \frac{1}{n} \), we derive a set of terms when substituting different values for \( n \). This particular sequence forms the starting point for a more profound exploration of its long-term behavior.
For an effective analysis, consider these aspects:

• Calculate individual terms clearly, focusing on what happens as \( n \) increases.
• Record how each successive term compares with the one before it—is there an evident trend or pattern?
• Determine any bounds or limits towards which the sequence terms seem to move.

In the given sequence, the decreasing portion \( \frac{1}{n} \) indicates that as \( n \) grows, \( a_n \) gets closer to 5. Examination of terms up to \( n = 10 \) already provides insight into this behavior, which paves the way for more detailed analytical predictions about the sequence's convergence properties.

Convergence and Divergence

Understanding whether a sequence converges or diverges is vital in sequence analysis. A sequence converges if its terms approach a specific finite number as \( n \) increases indefinitely. If not, and the terms either grow without bound or fluctuate without settling, the sequence diverges.

In our example sequence \( a_n = 5 - \frac{1}{n} \), as \( n \) becomes large, \( \frac{1}{n} \) becomes very small, causing \( a_n \) to approach the value 5. Hence, this sequence converges. When making such conjectures, look for:

• A trend where terms of the sequence get irresistibly close to a particular value.
• Analyze cases where as \( n \) increases, any potential deviations from a fixed point shrink.
• Consider mathematically proving your conjecture with a formal limit calculation, which in this case could be shown as: \( \lim_{{n \to \infty}} (5 - \frac{1}{n}) = 5 \).

Spotting the convergence or divergence of sequences is a fundamental skill that reveals the ultimate behavior of terms over their infinite progression. This exercise highlights how important visualization and symbolic analysis are in coming to these conclusions.