Problem 108 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdot \cdot \cdot-\frac{1}{128}$$

Step-by-Step Solution

Answer

The series can be written in the sigma notation as \(\sum_ {i=0}^{6} 1 \times (-\frac{1}{2})^i\) and the sum of the series is -2/3.

Step 1 Title

Identify the first term (a), common ratio (r), and number of terms (n). The first term a is 1. The common ratio r is -1/2 (each term is half of the preceding term and changes its sign). The number of terms n can be found by observing the pattern of the denominator (2^1, 2^2, 2^3...2^7) and associating it with n. So, n is 7.

Step 2 Title

Write the sum in sigma notation. This can be written as: \(\sum_ {i=0}^{n-1} a \times r^i = 1 \times (-\frac{1}{2})^0 + 1 \times (-\frac{1}{2})^1 + \cdots + 1 \times (-\frac{1}{2})^6\). This is because the series starts from the term with i=0 and ends at the term with \(i=n-1\) where \(n=7\).

Step 3 Title

Use a graphing utility or spreadsheet to compute the sum. Plug in the values of a, r and i in the formula for \(n-1\) terms with n=7. Now it can be computed as -2/3 with the help of any graphing utility or spreadsheet.

Key Concepts

Graphing Utility

In contemporary education and professional practice, the use of graphing utilities offers a significant advantage for visualizing and solving complex mathematical tasks. These utilities, which include calculators with graphing capabilities and software like spreadsheets, allow users to plot equations and view their graphical representations. When applied to series, these tools can perform rapid calculations that would be time-consuming by hand.
For the given exercise, the sigma notation can be translated into a format recognizable by the graphing utility, which calculates the sum quickly. Spreadsheet software, for instance, can use built-in functions to perform iterations of the series, sum them up, and even portray visual trends. Understanding how to operate these tools effectively can enhance a student's perspective on the role of mathematics in a digital context and equip them for real-world problem solving.

Convergence of Series

The concept of convergence is fundamental when dealing with series and sequences. A series is said to converge if the sum of its infinite terms approaches a finite limit. For a geometric series, this happens when the absolute value of the common ratio is less than 1.
In our specific example, we deal with a finite geometric series, so the idea of convergence isn't directly applicable. However, understanding convergence is essential when studying infinite series, as it informs whether meaningful sums can be derived. Convergence has practical implications in fields like computer science, physics, and economics, often used to predict outcomes or to check the feasibility of models over long-term projections.
For example, in infinite series calculations, the concept of convergence implies the behaviour of investments over time or the sum of infinite transactions. Students tackling these complex ideas often gain insights into how seemingly 'unbounded' mathematical entities can indeed exhibit bounded, predictable behaviour in many scenarios.