Problem 140 - 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 \(L ?\) $$ a_{1}=1, \quad a_{n+1}=a_{n}+(-2)^{n} $$

Step-by-Step Solution

Answer

The sequence diverges and does not have a limit.

Step 1: Calculate the first 25 terms

Start with the initial term, \( a_1 = 1 \). To generate the sequence, apply the recurrence relation \( a_{n+1} = a_n + (-2)^n \). The first few terms are computed as follows: - \( a_1 = 1 \) - \( a_2 = a_1 + (-2)^1 = 1 - 2 = -1 \) - \( a_3 = a_2 + (-2)^2 = -1 + 4 = 3 \) Continue this process to compute subsequent terms until you reach the 25th term.

Step 2: Plot the first 25 terms

Using a graphing tool or graphing software, plot the sequence values from \( a_1 \) to \( a_{25} \). Observe the trend of the sequence visually to determine whether or not it seems to be bounded.

Step 3: Analyze boundedness and convergence

Examine the plotted sequence. If the sequence values do not exceed a certain maximum or drop below a certain minimum consistently, it is bounded. Then, observe if the sequence approaches a specific value, indicating convergence, or if it oscillates indefinitely, indicating divergence.

Step 4: Determine if the sequence converges

Given the pattern of construction, since \((-2)^n\) rapidly increases or decreases with alternating signs for subsequent terms, the sequence does not approach a single value and hence, diverges. So, it doesn't have a limit \(L\).

Step 5: Consider conditions for convergence

If hypothetically, the sequence converged to a limit \(L\), one would determine \(N\) such that \( |a_n - L| \leq 0.01 \) or 0.0001. However, since the sequence diverges, this calculation is not applicable.

Key Concepts

Recurrence Relations

A recurrence relation is a way of defining a sequence based on previous terms. In simple terms, it's like a rule or formula that tells you how to get from one number in the sequence to the next. These relations are key in generating a sequence step by step. For example, in our sequence, the recurrence relation is \( a_{n+1} = a_n + (-2)^n \). This means that to find the next term, you take the current term and add \((-2)^n\) to it.

• Start with the initial value, which is often given, such as \( a_1 = 1 \).
• Use the recurrence relation repeatedly to find each subsequent term.
• This process continues to give us a sequence of numbers.

By carefully following the recurrence relation, sequences can be extended to as many terms as desired, enabling analysis and insights into their behavior.

Bounded Sequences

A sequence is considered bounded if it does not go beyond certain limits. In other words, there exists some maximum and minimum value that the sequence does not exceed. For instance, if no terms of a sequence are greater than a certain number, it is said to be bounded above. Conversely, if no terms are less than a certain number, it is bounded below.

• Bounded above: The sequence values do not exceed a particular value.
• Bounded below: The sequence values do not drop under a specific value.

In the exercise, plotting helps visually assess whether the sequence seems to consistently stay within upper or lower limits. This assessment is crucial in determining the potential for convergence as bounded sequences have a higher chance of converging than unbounded ones.

Convergence and Divergence

Understanding whether a series converges or diverges is essential in mathematical analysis. A sequence converges if its terms get increasingly close to a specific value as they progress to infinity. This specific value is called the limit. On the other hand, if a sequence doesn't approach any particular value, it diverges.

• Convergence: Approaching a specific limit over time.
• Divergence: Not settling to any value.

In the problem given, the sequence diverges. This is because the alternation in sign and the incrementing magnitude of \((-2)^n\) values keep the sequence from stabilizing to any single number. Identifying divergence or convergence assists in understanding the behavior and fate of sequences over time.

CAS (Computer Algebra Systems)

Computer Algebra Systems (CAS) are software programs that solve mathematical problems symbolically. These powerful tools are used to handle complex calculations that would be cumbersome by hand. In the context of sequences, CAS can compute numerous terms rapidly, plot sequences, and check boundedness or convergence effectively.

• Calculate terms efficiently and accurately.
• Plot sequences to visually interpret their behavior.
• Analyze boundedness and convergence with ease.

Using a CAS streamlines these processes, allowing us to focus more on interpreting the results rather than the mechanics of calculation, which enhances understanding and facilitates learning.