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Magazine Chapter 8: Problem 41
Use a formula to approximate the sum for \(n=4,7, \text { and } 10.\) $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\dots+\left(-\frac{1}{2}\right)^{n-1} $$
Short Answer
Expert verified
The approximate sums are \( \frac{5}{8} \), \( \frac{43}{64} \), and \( \frac{341}{512} \) for \( n = 4, 7, \) and \( 10 \) respectively.
Step by step solution
01
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 \( r = -\frac{1}{2} \).
02
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.
03
Substitute Values for \( n = 4 \)
Substitute \( n = 4 \), \( a = 1 \), and \( r = -\frac{1}{2} \) into the sum formula:\[ S_4 = 1 \cdot \frac{1-(-\frac{1}{2})^4}{1-(-\frac{1}{2})} = \frac{1-(\frac{1}{16})}{1+\frac{1}{2}} = \frac{1-\frac{1}{16}}{1.5} = \frac{15/16}{3/2} = \frac{15}{24} = \frac{5}{8} \]
04
Substitute Values for \( n = 7 \)
Substitute \( n = 7 \), \( a = 1 \), and \( r = -\frac{1}{2} \) into the sum formula:\[ S_7 = 1 \cdot \frac{1-(-\frac{1}{2})^7}{1-(-\frac{1}{2})} = \frac{1-(-\frac{1}{128})}{1+\frac{1}{2}} = \frac{1 + \frac{1}{128}}{1.5} = \frac{129/128}{3/2} = \frac{129}{192} = \frac{43}{64} \]
05
Substitute Values for \( n = 10 \)
Substitute \( n = 10 \), \( a = 1 \), and \( r = -\frac{1}{2} \) into the sum formula:\[ S_{10} = 1 \cdot \frac{1-(-\frac{1}{2})^{10}}{1-(-\frac{1}{2})} = \frac{1-(\frac{1}{1024})}{1+\frac{1}{2}} = \frac{1023/1024}{3/2} = \frac{1023}{1536} = \frac{341}{512} \]
06
Conclusion
The sums of the series for \( n = 4, 7, \) and \( 10 \) are \( \frac{5}{8} \), \( \frac{43}{64} \), and \( \frac{341}{512} \), respectively.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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.
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.
