Problem 148 - 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 \(L ?\) $$ a_{n}=\frac{n^{41}}{19^{n}} $$

Step-by-Step Solution

Answer

The sequence converges to 0. Find \(N\) where terms are ≤0.01 and 0.0001.

Step 1: Calculate First 25 Terms

We need to calculate the first 25 terms of the sequence given by \( a_n = \frac{n^{41}}{19^n} \). This can be achieved using a Computer Algebra System (CAS). For example, the first term \( a_1 = \frac{1^{41}}{19^1} = \frac{1}{19} \) and so forth up to \( a_{25} = \frac{25^{41}}{19^{25}} \). Evaluate these terms numerically to get an approximation.

Step 2: Plot the Terms

Generate a plot for the first 25 terms of the sequence using the CAS. On the x-axis, plot the term number \( n \) and on the y-axis, plot \( a_n \). Observe the behavior of the sequence visually from the plot.

Step 3: Determine Boundedness and Convergence

Examine the plot to determine if the sequence appears to be bounded above or below. Additionally, observe if the terms are approaching a particular value (convergence) or diverging. For this sequence, it appears to converge to zero because the terms diminish as \( n \) increases.

Step 4: Identify the Limit

Since the sequence converges, determine the limit \( L \) to which the sequence converges. From the observed behavior, we conjecture that \( L = 0 \).

Step 5: Find Integer N for \(|a_n - L| \leq 0.01\)

Calculate the smallest integer \( N \) such that \( |a_n - 0| \leq 0.01 \). This involves finding the point in the sequence where the absolute value of the terms is less than or equal to 0.01. Use the CAS to compute these terms numerically until the condition is satisfied.

Step 6: Terms Within 0.0001 of L

Next, determine how far you need to go in the sequence for \( |a_n - 0| \leq 0.0001 \). Calculate the terms using CAS until they are sufficiently small and note the smallest \( n \) for which this condition holds.

Key Concepts

Convergence and Divergence

In mathematics, understanding whether a sequence converges or diverges is crucial. A sequence converges if its terms approach a specific value as the sequence progresses. This value is called the limit. Conversely, a sequence diverges if the terms continue to increase or decrease without approaching any particular value. In the case of our sequence \( a_n = \frac{n^{41}}{19^n} \), convergence is observed. As \( n \) becomes very large, the rapidly increasing exponential in the denominator \( 19^n \) overpowers the polynomial growth in the numerator \( n^{41} \). This causes the terms to diminish towards zero. Hence, the sequence is converging to a limit \( L \), which in this example, is zero. Observing convergence or divergence helps us determine the long-term behavior of sequences, which can be essential in calculus and real-world applications.

CAS (Computer Algebra System)

A Computer Algebra System (CAS) is an incredibly powerful tool used in mathematics to perform operations on mathematical expressions. It's especially useful when dealing with complex sequences and series. By using a CAS, we can easily compute, visualize, and analyze sequences. In the exercise for sequence \( a_n = \frac{n^{41}}{19^n} \), the CAS allows us to calculate the first 25 terms with precision and create a plot to observe their behavior. These systems perform symbolic mathematics, which means they can handle equations exactly rather than approximately. This is particularly valuable for solving algebraic equations, simplifying expressions, and understanding the convergence and divergence of sequences. In educational settings, CAS tools help students explore mathematical concepts more deeply by instantly providing feedback and supporting interactive learning.

Limits of Sequences

The limit of a sequence is the value the terms approach as the index number increases indefinitely. In the sequence \( a_n = \frac{n^{41}}{19^n} \), we examined whether the terms are getting closer to some value as \( n \) increases without bound. We hypothesized that our sequence converges to zero by observing the diminishing size of the terms \( a_n \). Calculating the limit involves seeking this behavior analytically or using computational tools. For sequences that converge, we can find integer \( N \) such that for all \( n \geq N \), the terms are within a certain distance from the limit \( L \). This exercise also asks for \( |a_n - L| \leq 0.01 \) and \( \leq 0.0001 \), indicating how close the sequence terms are to the limit for sufficiently large \( n \). Overall, the concept of limits is fundamental in understanding sequence behavior and is a cornerstone in calculus and analysis, revealing insights into functions and their continuity.