Problem 90 - Essential Calculus

Question

Use a graphing utility to determine the first term that is less than 0.0001 in each of the convergent series. Note that the answers are very different. Explain how this will affect the rate at which each series converges. $$ \sum_{n=1}^{\infty} \frac{1}{2^{n}}, \quad \sum_{n=1}^{\infty}(0.01)^{n} $$

Step-by-Step Solution

Answer

For the first series, the 14th term is the first one less than 0.0001. For the second series, the 4th term is the first one less than 0.0001. This difference shows that the second series, with a smaller ratio (0.01 compared to 0.5), converges at a slower pace than the first series.

Step 1: Understand the individual series

First, analyze the given series separately. The first series is a geometric series with a common ratio of \( \frac{1}{2} \). The second series is also a geometric series with a common ratio of 0.01.

Step 2: Calculate when the term is less than 0.0001

Next, set up an inequality to determine when the term of the series is less than \(0.0001\). In the first case, solve the inequality \( \frac{1}{2^{n}} < 0.0001 \) by natural logarithms and in the second case solve \( (0.01)^{n} < 0.0001 \) likewise with logarithms.

Step 3: Use a graphing utility to visualize

Input the sequences into a graphing utility software, where 'x' represents the term number 'n'. Trace along the graph to locate which term number the graph initially drops below the line y = 0.0001 for both the series. Note the 'x'-coordinate of these point, that is the term number you are looking for.

Step 4: Understand the implications on rate of convergence

The series with the smaller ratio will converge more slowly than the one with a larger ratio. This conclusion can be drawn by comparing the two series and observing which one reached the term value less than 0.0001 first.

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, in the series \( \frac{1}{2^n} \) from the exercise, every term is half of the preceding term, making \( \frac{1}{2} \) the common ratio. This type of series can be summed to a finite value when the absolute value of the common ratio is less than one, as seen here.

Geometric series have a critical application in fields such as finance for calculating compound interest, in computer science for analysis of algorithms' efficiency, and in physics for series circuits or wave forms, among others. Recognizing a geometric series allows mathematicians and students alike to apply formulas for the sum of an infinite series when |r| < 1:

Rate of Convergence

The rate of convergence refers to how quickly the sequence of partial sums of a series approaches its limit as the number of terms increases. In the context of the given exercise, we are exploring two different geometric series, \( \frac{1}{2^n} \) and \( (0.01)^n \) to determine the first term that is less than 0.0001. The former, with a common ratio of \( \frac{1}{2} \) has a much faster rate of convergence compared to the latter, which has a common ratio of 0.01.

Understanding the rate of convergence is fundamental when working with series, as it informs us about the efficiency and practicality of using the series to approximate solutions in various disciplines such as mathematics, economics, and computer science. In numerical analysis, faster convergence can lead to more accurate results in fewer steps. The concept of rate of convergence is essential for error analysis and ensuring the reliability of numerical computations.

Graphing Utility

A graphing utility is an indispensable tool in mathematics for visualizing functions and sequences - particularly useful when assessing series convergence. The exercise requires students to use a graphing utility to locate visually where the term of the series is less than 0.0001 for both \( \frac{1}{2^n} \) and \( (0.01)^n \) series. By entering the sequence into a graphing program and plotting the terms against their index number, students can see where the graph falls below a specified threshold.

Graphing utilities assist not only in finding specific numerical answers but also in understanding the behavior of functions and series over intervals. They often provide additional features like zooming, tracing, or computing derivatives and integrals, which enhance comprehension. In the classroom or for self-study, incorporating graphing utilities into the learning process can significantly improve the grasp of abstract concepts and the intuition behind mathematical analyses.