Understanding The Series Representation Of Sum From N=0 To 5 Of (1/3)(-2)^n
Write the series represented by the summation from n=0 to 5 of (1/3)(-2)^n.
Introduction
In the realm of mathematics, series play a crucial role in representing and understanding various mathematical concepts. A series is essentially the sum of the terms of a sequence. This article delves into the specific series represented by the summation $\sum_{n=0}^5 \frac{1}{3}(-2)^n$, providing a detailed breakdown of its terms, calculation, and underlying mathematical principles. Understanding series like this is fundamental in various areas of mathematics, including calculus, analysis, and numerical methods. We will explore how to expand this summation, identify its pattern, and ultimately calculate its value. The knowledge gained from this exploration will provide a solid foundation for understanding more complex series and their applications in diverse fields.
Understanding the Summation Notation
The summation notation, represented by the Greek letter sigma (∑), provides a concise way to express the sum of a series. In the given expression, $\sum_{n=0}^5 \frac{1}{3}(-2)^n$, the notation indicates that we need to sum the terms generated by the expression ${\frac{1}{3}(-2)^n}$ as ${n}$ varies from 0 to 5. The variable ${n}$ is the index of summation, and the numbers 0 and 5 define the lower and upper limits of the summation, respectively. This notation is a powerful tool for representing series in a compact and understandable manner. By understanding the components of the summation notation, we can effectively translate it into an expanded form, which allows us to calculate the sum of the series. This process involves substituting each integer value of ${n}$ within the specified range into the expression and then adding the resulting terms together. The summation notation not only simplifies the representation of series but also provides a clear and unambiguous way to define the terms that need to be included in the sum. Mastering this notation is essential for working with series and sequences in mathematics.
Expanding the Series Term by Term
To fully grasp the series, let's expand it term by term. This involves substituting the values of $n}$ from 0 to 5 into the expression ${\frac{1}{3}(-2)^n}$ and writing out each term. This process will reveal the pattern of the series and make it easier to calculate the sum. The first term is obtained when ${n=0}$3}(-2)^0 = \frac{1}{3}(1) = \frac{1}{3}}. The second term is when \${n=1\}{3}(-2)^1 = \frac{1}{3}(-2) = -\frac{2}{3}}$. Continuing this pattern, we get the following terms:
- $n=2}${3}(-2)^2 = \frac{1}{3}(4) = \frac{4}{3}}$
- $n=3}${3}(-2)^3 = \frac{1}{3}(-8) = -\frac{8}{3}}$
- $n=4}${3}(-2)^4 = \frac{1}{3}(16) = \frac{16}{3}}$
- $n=5}${3}(-2)^5 = \frac{1}{3}(-32) = -\frac{32}{3}}$
Therefore, the expanded form of the series is: $\frac{1}{3} - \frac{2}{3} + \frac{4}{3} - \frac{8}{3} + \frac{16}{3} - \frac{32}{3}$. By writing out the series in this expanded form, we can clearly see the alternating signs and the geometric progression of the terms. This step is crucial for understanding the nature of the series and preparing for the calculation of its sum. The process of expanding the series term by term not only aids in visualization but also helps in identifying any patterns or relationships between the terms, which can be beneficial in further analysis.
Identifying the Series Type: Geometric Series
Upon examining the expanded series, it becomes evident that it is a geometric series. A geometric series is a series in which the ratio between consecutive terms is constant. This constant ratio is known as the common ratio (${r}). In our series, the ratio between any two consecutive terms is -2. For instance, (-2/3) / (1/3) = -2, (4/3) / (-2/3) = -2, and so on. Identifying the series as geometric allows us to utilize specific formulas and techniques to calculate its sum. The general form of a geometric series is a + ar + ar^2 + ar^3 + ..., where \${a\} is the first term and ${r}$ is the common ratio. In our case, the first term ${a}$ is 1/3, and the common ratio ${r}$ is -2. Recognizing the geometric nature of the series is a key step in efficiently determining its sum. This identification allows us to apply the formula for the sum of a finite geometric series, which simplifies the calculation process significantly. The ability to recognize different types of series, such as geometric, arithmetic, or harmonic, is a fundamental skill in mathematics, enabling us to apply the appropriate methods for analysis and computation.
Calculating the Sum of the Geometric Series
Since we have identified the series as a geometric series, we can use the formula for the sum of a finite geometric series to calculate its value. The formula for the sum of the first $n}$ terms of a geometric series is given by3}}, \${r = -2\}, and ${n = 6}$ (since we are summing from ${n=0}$ to ${n=5}$, which includes 6 terms). Plugging these values into the formula, we get3}(1 - (-2)^6) / (1 - (-2)). Simplifying this expression, we have{3}(1 - 64) / (1 + 2) = \frac{1}{3}(-63) / 3 = -\frac{63}{9} = -7. Therefore, the sum of the series $\sum_{n=0}^5 \frac{1}{3}(-2)^n$ is -7. This calculation demonstrates the efficiency of using the formula for the sum of a finite geometric series. By correctly identifying the series type and applying the appropriate formula, we can quickly and accurately determine the sum. Understanding and applying these formulas is crucial for solving problems involving series and sequences in mathematics.
Detailed Step-by-Step Calculation
To further clarify the calculation, let's break it down step-by-step:
- Identify the parameters:
- First term ($a}$){3}}$
- Common ratio (${r}$): -2
- Number of terms (${n}): 6 (from \${n=0\} to ${n=5}$)
- Apply the formula for the sum of a finite geometric series:
- ${S_n = a(1 - r^n) / (1 - r)}$
- ${S_6 = \frac{1}{3}(1 - (-2)^6) / (1 - (-2))}$
- Calculate (-2)^6:
- ${(-2)^6 = 64}$
- Substitute the value back into the formula:
- ${S_6 = \frac{1}{3}(1 - 64) / (1 + 2)}$
- Simplify the numerator:
- ${1 - 64 = -63}$
- ${S_6 = \frac{1}{3}(-63) / 3}$
- Multiply ${\frac{1}{3}}$ by -63:
- ${\frac{1}{3}(-63) = -21}$
- ${S_6 = -21 / 3}$
- Divide -21 by 3:
- ${S_6 = -7}$
This step-by-step breakdown illustrates the application of the formula and the arithmetic operations involved in calculating the sum of the series. Each step is clearly defined, making it easier to follow the process and understand the reasoning behind each calculation. This detailed approach is particularly helpful for those who are new to the concept of geometric series and want to gain a thorough understanding of the calculation process. By breaking down the problem into smaller, manageable steps, we can avoid errors and ensure accurate results.
Conclusion: The Sum of the Series
In conclusion, by expanding the series, identifying it as a geometric series, and applying the appropriate formula, we have determined that the sum of the series $\sum_{n=0}^5 \frac{1}{3}(-2)^n$ is -7. This exercise demonstrates the importance of understanding summation notation, recognizing series types, and utilizing relevant formulas to calculate sums. The concept of geometric series is fundamental in mathematics and has applications in various fields, including finance, physics, and computer science. Understanding how to work with series and calculate their sums is a crucial skill for anyone pursuing further studies in mathematics or related disciplines. This article has provided a comprehensive exploration of the series, from its initial representation to its final calculated sum, offering a clear and detailed explanation of each step involved. By mastering these concepts, readers can confidently tackle similar problems and expand their understanding of series and sequences.
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