Chapter 12: Q 66. (page 809)

In Problems 61–70, express each sum using summation notation.
$\frac{2}{3} - \frac{4}{9} + \frac{8}{27} - . . . . . . . . + {(- 1)}^{12} {(\frac{2}{3})}^{11}$

Short Answer

Expert verified

The required summation notation is $\frac{2}{3} - \frac{4}{9} + \frac{8}{27} - . . . . . . . . + {(- 1)}^{12} {(\frac{2}{3})}^{11} = \sum_{k = 1}^{11} {(- 1)}^{n + 1} {(\frac{2}{3})}^{k}$ .

Step by step solution

Step 1. Write the given information.

The given sum is:

\frac{2}{3} - \frac{4}{9} + \frac{8}{27} - . . . . . . . . + {(- 1)}^{12} {(\frac{2}{3})}^{11}

Step 2. Write the sum using summation notation.

Write each term in the same format:

$\frac{2}{3} - \frac{4}{9} + \frac{8}{27} - . . . . . . . . + {(- 1)}^{12} {(\frac{2}{3})}^{11} = {(- 1)}^{2} {(\frac{2}{3})}^{1} + {(- 1)}^{3} {(\frac{2}{3})}^{2} + {(- 1)}^{4} {(\frac{2}{3})}^{3} + . . . . . . . + {(- 1)}^{12} {(\frac{2}{3})}^{11} = {(- 1)}^{1 + 1} {(\frac{2}{3})}^{1} + {(- 1)}^{2 + 1} {(\frac{2}{3})}^{2} + {(- 1)}^{3 + 1} {(\frac{2}{3})}^{3} + . . . . . . . + {(- 1)}^{11 + 1} {(\frac{2}{3})}^{11} = \sum_{k = 1}^{11} {(- 1)}^{n + 1} {(\frac{2}{3})}^{k}$

Therefore,role="math" localid="1646798011820" $\frac{2}{3} - \frac{4}{9} + \frac{8}{27} - . . . . . . . . + {(- 1)}^{12} {(\frac{2}{3})}^{11} = \sum_{k = 1}^{11} {(- 1)}^{n + 1} {(\frac{2}{3})}^{k}$ .

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In Problems 61–70, express each sum using summation notation.
$\frac{2}{3} - \frac{4}{9} + \frac{8}{27} - . . . . . . . . + {(- 1)}^{12} {(\frac{2}{3})}^{11}$

Short Answer

Step by step solution

Step 1. Write the given information.

Step 2. Write the sum using summation notation.

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In Problems 61–70, express each sum using summation notation.23-49+827-........+-1122311

Short Answer

Step by step solution

Step 1. Write the given information.

Step 2. Write the sum using summation notation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Decision Maths

Logic and Functions

Theoretical and Mathematical Physics

Geometry

Pure Maths

Study anywhere. Anytime. Across all devices.

In Problems 61–70, express each sum using summation notation.
$\frac{2}{3} - \frac{4}{9} + \frac{8}{27} - . . . . . . . . + {(- 1)}^{12} {(\frac{2}{3})}^{11}$