Chapter 13: Q. 4 (page 1003)

$Express the sum \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} using double - summation notation .$

Short Answer

Expert verified

$\frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} = \sum_{j = 1}^{j = 2} \sum_{k = 5}^{k = 8} \frac{2^{j}}{k}$

Step by step solution

Step 1. Given Information

$\frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} in double - summation notation .$

Step 2. Finding common terms and converting to summation

$\frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} = (\frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8}) + (\frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8}) = (\frac{2^{1}}{5} + \frac{2^{1}}{6} + \frac{2^{1}}{7} + \frac{2^{1}}{8}) + (\frac{2^{2}}{5} + \frac{2^{2}}{6} + \frac{2^{2}}{7} + \frac{2^{2}}{8}) The common term is \frac{2^{j}}{5} + \frac{2^{j}}{6} + \frac{2^{j}}{7} + \frac{2^{j}}{8} and ranges from 1 to 2, which can be converted as \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} = (\frac{2^{1}}{5} + \frac{2^{1}}{6} + \frac{2^{1}}{7} + \frac{2^{1}}{8}) + (\frac{2^{2}}{5} + \frac{2^{2}}{6} + \frac{2^{2}}{7} + \frac{2^{2}}{8}) = \sum_{j = 1}^{2} \frac{2^{j}}{5} + \frac{2^{j}}{6} + \frac{2^{j}}{7} + \frac{2^{j}}{8} The common term is \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} and ranges from 5 to 8, which can be converted as \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} = \sum_{j = 1}^{2} \frac{2^{j}}{5} + \frac{2^{j}}{6} + \frac{2^{j}}{7} + \frac{2^{j}}{8} = \sum_{j = 1}^{j = 2} \sum_{k = 5}^{k = 8} \frac{2^{j}}{k} Hence, \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} = \sum_{j = 1}^{j = 2} \sum_{k = 5}^{k = 8} \frac{2^{j}}{k}$

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$Express the sum \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} using double - summation notation .$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding common terms and converting to summation

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Expressthesum25+26+27+28+45+46+47+48usingdouble-summationnotation.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding common terms and converting to summation

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Decision Maths

Theoretical and Mathematical Physics

Mechanics Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

$Express the sum \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} using double - summation notation .$