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Write the series using summation notation. ++或 1 1 +... 81 i=1

Write the series using summation notation. ++或 1 1 +... 81 i=1

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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what is the summation notation of the series?

### Series and Summation Notation To represent the following series in summation notation: \[ \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \ldots \] we first observe the pattern in the terms of the series. Each term is a power of \(\frac{1}{3}\): 1. \(\frac{1}{3} = \left( \frac{1}{3} \right)^1\) 2. \(\frac{1}{9} = \left( \frac{1}{3} \right)^2\) 3. \(\frac{1}{27} = \left( \frac{1}{3} \right)^3\) 4. \(\frac{1}{81} = \left( \frac{1}{3} \right)^4\) 5. And so on... Thus, the \(i\)-th term of this series is \(\left( \frac{1}{3} \right)^i\). Next, we use summation notation to express this series. The general formula for summation notation is: \[ \sum_{i=a}^{b} f(i) \] where \(i\) is the index of summation, \(a\) is the lower limit, \(b\) is the upper limit, and \(f(i)\) is the function of \(i\) that gives the terms of the series. For this infinite series, we start from \(i = 1\) and proceed indefinitely. Hence, the summation notation for the given series is: \[ \sum_{i=1}^{\infty} \left( \frac{1}{3} \right)^i \] This is visually represented in the image as follows: - There is a sigma symbol \( \sum \) indicating summation. - The index \(i=1\) is written below the sigma symbol, representing the starting value of the summation index. - The function to be summed \(\left( \frac{1}{3} \right)^i\) is written to the right of the sigma symbol. The final notation would be: \[ \sum_{i=1}^{\infty} \left( \frac{1}{3} \right)^i \] This summarizes an infinite geometric series where each term
Transcribed Image Text:### Series and Summation Notation To represent the following series in summation notation: \[ \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \ldots \] we first observe the pattern in the terms of the series. Each term is a power of \(\frac{1}{3}\): 1. \(\frac{1}{3} = \left( \frac{1}{3} \right)^1\) 2. \(\frac{1}{9} = \left( \frac{1}{3} \right)^2\) 3. \(\frac{1}{27} = \left( \frac{1}{3} \right)^3\) 4. \(\frac{1}{81} = \left( \frac{1}{3} \right)^4\) 5. And so on... Thus, the \(i\)-th term of this series is \(\left( \frac{1}{3} \right)^i\). Next, we use summation notation to express this series. The general formula for summation notation is: \[ \sum_{i=a}^{b} f(i) \] where \(i\) is the index of summation, \(a\) is the lower limit, \(b\) is the upper limit, and \(f(i)\) is the function of \(i\) that gives the terms of the series. For this infinite series, we start from \(i = 1\) and proceed indefinitely. Hence, the summation notation for the given series is: \[ \sum_{i=1}^{\infty} \left( \frac{1}{3} \right)^i \] This is visually represented in the image as follows: - There is a sigma symbol \( \sum \) indicating summation. - The index \(i=1\) is written below the sigma symbol, representing the starting value of the summation index. - The function to be summed \(\left( \frac{1}{3} \right)^i\) is written to the right of the sigma symbol. The final notation would be: \[ \sum_{i=1}^{\infty} \left( \frac{1}{3} \right)^i \] This summarizes an infinite geometric series where each term
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