Problem 147 - Thomas Calculus

Question

Use a CAS to perform the following steps for the sequences in Exercises \(137-148 .\) a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit L? b. If the sequence converges, find an integer \(N\) such that \(\quad\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$ a_{n}=\frac{8^{n}}{n !} $$

Step-by-Step Solution

Answer

The sequence is bounded below and converges to 0. For \(|a_n| \leq 0.01\), \(N = 9\); for \(|a_n| \leq 0.0001\), \(N = 13\).

Step 1: Calculate the Sequence

Calculate the first 25 terms of the sequence, where the sequence is given as \( a_n = \frac{8^n}{n!} \). Here, \( n! \) denotes the factorial of \( n \) and \( 8^n \) is 8 raised to the power of \( n \).

Step 2: Plot the Sequence

Using a computational tool (CAS), plot the first 25 terms of the sequence. Observe the graph to understand the behavior of the sequence visually.

Step 3: Analyze Boundedness

Examine the plotted graph to determine if the sequence appears to be bounded from above or below. Look for any horizontal asymptotes or if the sequence's growth is limited.

Step 4: Determine Convergence or Divergence

From the plot and calculated terms, assess if the sequence is converging to a specific number or diverging. A converging sequence will approach a finite limit, while a diverging sequence will not approach any specific value.

Step 5: Calculate the Limit

If the sequence converges, use the plot and numerical evidence to estimate the limit \( L \). For factorial and exponential functions, it is often observed that \( \frac{8^n}{n!} \) converges to 0 as \( n \to \infty \).

Step 6: Find N for 0.01 Criteria

If the sequence converges, calculate \( N \) such that \( |a_n - L| \leq 0.01 \) for \( n \geq N \), with \( L \) being the estimated limit (0 in our case). Check terms for where this condition holds.

Step 7: Find N for 0.0001 Criteria

Similarly, find the smallest \( N \) such that \( |a_n - L| \leq 0.0001 \) for \( n \geq N \). This will indicate how far in the sequence you must go for terms to be within 0.0001 of 0.

Key Concepts

Bounded Sequences

In mathematics, a sequence is simply an ordered list of numbers. A sequence is termed "bounded" if its values don’t go beyond certain limits, either above, below, or both. A sequence is bounded above if there is a real number, say M, such that every term of the sequence is less than or equal to M. Similarly, it's bounded below if there’s a number, say m, where every term is greater than or equal to m.
In the study of sequences, boundedness helps us understand the constraints and predictability of the sequence. When dealing with complex expressions, such as the factorial and exponential functions combined in the sequence formula given in the exercise: \[ a_n = \frac{8^n}{n!} \] It might not be immediately obvious whether a sequence is bounded. However, computational and graphical analyses reveal the pattern of such sequences. If, through observation, the terms of the sequence do not exceed a particular value, we can say it is bounded from above. In our case, a sequence like this tends toward zero as given n grows, indicating its bounded nature.

Computational Tools (CAS)

Computational tools, like Computer Algebra Systems (CAS), are a game-changer in the world of mathematics. They take the computational burden off students and researchers, allowing more room for critical thinking and understanding. CAS can handle various mathematical tasks, including plotting sequences, like the one in the given exercise, where you need to plot the first 25 terms of the sequence \( a_n = \frac{8^n}{n!} \).
Using a CAS for plotting helps you visually evaluate the behavior of the sequence. Graphs plotted with CAS can show whether the sequence's terms grow towards infinity or compress towards a certain limit. With this information, evaluating convergence or divergence becomes more intuitive. Observations from CAS plots can support calculations, offering a visual complementary insight into complex problems.

Factorial and Exponential Functions

Factorial and exponential functions can produce rapidly growing or decaying sequences. Understanding these functions' nature is crucial when analyzing sequences such as the exercise's sequence \( a_n = \frac{8^n}{n!} \).
#### Factorial Function The factorial of a number \( n \), denoted \( n! \), is the product of all positive integers up to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). The factorial grows extremely fast, especially when compared to polynomial or logarithmic growth.
#### Exponential Function An exponential function such as \( 8^n \) involves raising a constant base to a variable exponent. Exponential growth can outpace linear functions rapidly, but when a factorial function is in the denominator, it overtakes the exponential at large values of \( n \).
This combination means that for large \( n \), the factorial in the denominator grows faster than the exponential in the numerator, ultimately leading the sequence to tend towards zero. This is why the sequence \( a_n = \frac{8^n}{n!} \) is convergent, and its limit is zero.