Problem 145 - 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 \(L ?\) $$ a_{n}=(0.9999)^{n} $$

Step-by-Step Solution

Answer

The sequence converges to 0. For terms within 0.01 of 0, \( N = 461 \); for 0.0001, \( n = 9211 \).

Step 1: Calculate the First 25 Terms

Calculate the first 25 terms of the sequence using the formula \( a_n = (0.9999)^n \). We find:- \( a_1 = (0.9999)^1 = 0.9999 \)- \( a_2 = (0.9999)^2 \approx 0.9998 \)- Continue this calculation up to \( a_{25} \). Every term gradually approaches 0.

Step 2: Plot the First 25 Terms

Plot the calculated terms \( a_1, a_2, ..., a_{25} \) on a graph. This will visually show how each term is getting closer to 0 as \( n \) increases.

Step 3: Determine Boundedness and Convergence

Observe the plot and calculated terms:- The terms are bounded above by 1 (since \( (0.9999)^n < 1 \) for all \( n \)).- The terms also approach 0, which suggests the sequence is bounded below by 0.- As \( n \to \infty \), \( a_n \to 0 \). Thus, the sequence converges to a limit \( L = 0 \).

Step 4: Find N for the Given Condition

Since the sequence converges to \( L = 0 \), find the smallest integer \( N \) such that \( |a_n - 0| \leq 0.01 \).- Solve \( (0.9999)^n \leq 0.01 \).- Taking logarithms gives \( n \geq \frac{\log(0.01)}{\log(0.9999)} \approx 460.52 \).- Round up to find \( N = 461 \).

Step 5: Determine Terms Within 0.0001 of Limit

Similarly, for \( |a_n - 0| \leq 0.0001 \), solve \( (0.9999)^n \leq 0.0001 \).- Taking logarithms gives \( n \geq \frac{\log(0.0001)}{\log(0.9999)} \approx 9210.75 \).- Round up to find \( n = 9211 \).

Key Concepts

Bounded Sequences

In the world of sequences, understanding if a sequence is bounded is crucial. A sequence is said to be bounded if there is a real number that serves as a ceiling and/or floor for all terms in the sequence. Basically, no term should exceed the upper bound, and no term should fall below the lower bound. In the context of the sequence given by the formula \(a_n = (0.9999)^n\), this sequence is clearly bounded.

- **Upper Bound**: Each term \( (0.9999)^n \) is less than 1, making 1 the upper bound.
- **Lower Bound**: As \( n \) increases, \((0.9999)^n\) approaches zero. Therefore, 0 is a natural lower bound.

Identifying these bounds helps us determine the limits within which the sequence operates, aiding in further analysis and understanding of its behavior.

Limit of a Sequence

The limit of a sequence describes what the sequence approaches as the term number grows indefinitely. If this limit exists, the sequence is said to converge. Otherwise, it diverges. For our sequence \(a_n = (0.9999)^n\), we can see it converges to a specific value, here noted as \( L = 0 \).

Why does it converge to 0?

• Each term \((0.9999)^n\) gets smaller as \(n\) increases.
• The multiplier 0.9999 is less than 1, continuously reducing the value of future terms.

Verifying sequence convergence, as we did, can be pivotal, as it provides a sense of predictability and control over the behavior as terms extend towards infinity.

Calculus Sequences

Sequences in calculus are fundamental constructs that lead us to bigger concepts like series and functions. Through calculus, we grasp how sequences behave under infinity, boundedness, and limit conditions. The sequence, \(a_n\), explained in this example, is a gateway to understanding how sequences are used in calculus.

Calculus provides the tools to analyze \(a_n = (0.9999)^n\), clearly showing:

• **Convergence**: Using logarithmic properties to find when the terms are near to the limit.
• **Precision**: Calculating the precise point \(N\) at which terms reach within a degree of closeness to the limit.

Recognizing these aspects aids in mastering calculus sequences, helping students predict behaviors of more complex sequences, and allowing application to real-world scenarios in mathematics and science.