Problem 41 - College Algebra with Modeling and Visualization

Question

Use a formula to approximate the sum for \(n=4,7, \text { and } 10.\) $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\dots+\left(-\frac{1}{2}\right)^{n-1} $$

Step-by-Step Solution

Answer

The approximate sums are \( \frac{5}{8} \), \( \frac{43}{64} \), and \( \frac{341}{512} \) for \( n = 4, 7, \) and \( 10 \) respectively.

Step 1: Identify the Type of Series

The series given is an alternating geometric series. The general form of a geometric series is \( a + ar + ar^2 + \ldots + ar^{n-1} \), where \( a \) is the first term and \( r \) is the common ratio. In our case, \( a = 1 \) and \( r = -\frac{1}{2} \).

Step 2: Write the Formula for the Sum of the Series

The sum \( S_n \) of the first \( n \) terms of a geometric series can be calculated by the formula: \[ S_n = a \frac{1-r^n}{1-r} \] where \( a = 1 \) and \( r = -\frac{1}{2} \) in this scenario.

Step 3: Substitute Values for \( n = 4 \)

Substitute \( n = 4 \), \( a = 1 \), and \( r = -\frac{1}{2} \) into the sum formula:\[ S_4 = 1 \cdot \frac{1-(-\frac{1}{2})^4}{1-(-\frac{1}{2})} = \frac{1-(\frac{1}{16})}{1+\frac{1}{2}} = \frac{1-\frac{1}{16}}{1.5} = \frac{15/16}{3/2} = \frac{15}{24} = \frac{5}{8} \]

Step 4: Substitute Values for \( n = 7 \)

Substitute \( n = 7 \), \( a = 1 \), and \( r = -\frac{1}{2} \) into the sum formula:\[ S_7 = 1 \cdot \frac{1-(-\frac{1}{2})^7}{1-(-\frac{1}{2})} = \frac{1-(-\frac{1}{128})}{1+\frac{1}{2}} = \frac{1 + \frac{1}{128}}{1.5} = \frac{129/128}{3/2} = \frac{129}{192} = \frac{43}{64} \]

Step 5: Substitute Values for \( n = 10 \)

Substitute \( n = 10 \), \( a = 1 \), and \( r = -\frac{1}{2} \) into the sum formula:\[ S_{10} = 1 \cdot \frac{1-(-\frac{1}{2})^{10}}{1-(-\frac{1}{2})} = \frac{1-(\frac{1}{1024})}{1+\frac{1}{2}} = \frac{1023/1024}{3/2} = \frac{1023}{1536} = \frac{341}{512} \]

Step 6: Conclusion

The sums of the series for \( n = 4, 7, \) and \( 10 \) are \( \frac{5}{8} \), \( \frac{43}{64} \), and \( \frac{341}{512} \), respectively.

Key Concepts

Alternating Series

An alternating series is a type of series where the signs of its terms alternate between positive and negative. This kind of series often takes the form: \( a - b + c - d + \ldots \). Alternating series can converge to a sum, depending on the pattern and size of the terms. In mathematics, especially when dealing with functions, alternating series are important because they can help to simplify complicated calculations.

• For each term in an alternating series, the sign changes from positive to negative or vice versa.
• In financial mathematics and physics, alternating series allow modeling of certain behaviors such as oscillations and wave patterns.

In the given exercise, our alternating series is geometric with a first term \( a = 1 \) and a common ratio \( r = -\frac{1}{2} \). The alternating nature comes from the negative common ratio causing the terms to flip signs.

Sum Formula

Finding the sum of a geometric series helps us analyze and predict patterns within standard sets of numbers. A geometric series is defined by its first term, \( a \), and a common ratio, \( r \). The sum of the first \( n \) terms of a geometric series is calculated using:\[S_n = a \frac{1-r^n}{1-r}\]This formula comes in handy when the number of terms, the first term, and the common ratio are known.

• \( a \) is the initial term of the series.
• \( r \) is the common ratio, which is the ratio of any term to its preceding term.
• The expression \( r^n \) represents the \( n \)-th power of \( r \), and influences how each subsequent term affects the series' sum.

In our exercise, using the sum formula with \( n = 4, 7, \text{ and } 10 \) gave us approximations for the sum of the series like \( \frac{5}{8} \), \( \frac{43}{64} \), and \( \frac{341}{512} \).

Series Approximation

Approximation in series is all about finding a sum that is close to the true sum of an infinite series by calculating only a finite number of terms. This concept is critical when the entire series either diverges to infinity or becomes difficult to calculate entirely. By selecting a finite number of terms, such as \( n = 4, 7, \) or \( 10 \), we can estimate a result to a reasonable degree of accuracy.

• Approximations help manage computations and are especially useful in engineering and computer simulations where exact precision might not be feasible.
• The approximation usually improves as more terms are included in the sum.

In practice, you often choose \( n \) based on the level of precision required for your task. For the given series, choosing different \( n \) helps to understand how close the approximation is to the eventual limit.

College Algebra

College algebra involves understanding and applying algebraic concepts, often serving as the foundational skills for higher-level mathematics courses. Topics typically include equations, functions, polynomial operations, and series which prepare students for calculus and other advanced topics.

• It helps in developing problem-solving skills that are applicable across various scientific fields.
• Algebraic series and approximations provide practical knowledge for real-world problems such as interest computations or analyzing periodic functions.

In college algebra, understanding both alternating and geometric series enhances your ability to deal with complex equations and mathematical models. It prepares you for more advanced studies in calculus, where series become even more significant in deriving and solving equations.