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3. Σ (i-1) and Σ -k). | k=1

3. Σ (i-1) and Σ -k). | k=1

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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How would you state whether the given sums are equal or unequal #3?

### Mathematical Summation Notation **3.** - \(\sum_{i=1}^{10} i(i-1)\) and \(\sum_{k=1}^{10} (k^2 - k)\). **4.** - \(\sum_{i=1}^{8} i^3\) and \(\sum_{i=1}^{4} i^3 + \sum_{i=5}^{8} i^3\). ### Explanation: **Notation Overview:** - The symbol \(\sum\) represents a summation, which is the addition of a sequence of numbers. - The index \(i\) or \(k\) is the variable of summation. - The numbers below and above the summation symbol indicate the starting and ending values for the index. **Detailed Explanation:** **3.** - The first expression, \(\sum_{i=1}^{10} i(i-1)\), calculates the sum of the product of \(i\) and \((i-1)\) from \(i=1\) to \(i=10\). - The second expression, \(\sum_{k=1}^{10} (k^2 - k)\), calculates the sum of \((k^2 - k)\) from \(k=1\) to \(k=10\). **4.** - \(\sum_{i=1}^{8} i^3\) calculates the sum of the cubes of \(i\) from \(i=1\) to \(i=8\). - The expression \(\sum_{i=1}^{4} i^3 + \sum_{i=5}^{8} i^3\) is equivalent to \(\sum_{i=1}^{8} i^3\), as it divides the summation into two parts: from \(i=1\) to \(i=4\), and from \(i=5\) to \(i=8\).
Transcribed Image Text:### Mathematical Summation Notation **3.** - \(\sum_{i=1}^{10} i(i-1)\) and \(\sum_{k=1}^{10} (k^2 - k)\). **4.** - \(\sum_{i=1}^{8} i^3\) and \(\sum_{i=1}^{4} i^3 + \sum_{i=5}^{8} i^3\). ### Explanation: **Notation Overview:** - The symbol \(\sum\) represents a summation, which is the addition of a sequence of numbers. - The index \(i\) or \(k\) is the variable of summation. - The numbers below and above the summation symbol indicate the starting and ending values for the index. **Detailed Explanation:** **3.** - The first expression, \(\sum_{i=1}^{10} i(i-1)\), calculates the sum of the product of \(i\) and \((i-1)\) from \(i=1\) to \(i=10\). - The second expression, \(\sum_{k=1}^{10} (k^2 - k)\), calculates the sum of \((k^2 - k)\) from \(k=1\) to \(k=10\). **4.** - \(\sum_{i=1}^{8} i^3\) calculates the sum of the cubes of \(i\) from \(i=1\) to \(i=8\). - The expression \(\sum_{i=1}^{4} i^3 + \sum_{i=5}^{8} i^3\) is equivalent to \(\sum_{i=1}^{8} i^3\), as it divides the summation into two parts: from \(i=1\) to \(i=4\), and from \(i=5\) to \(i=8\).
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