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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![**Topic: Sigma Notation in Mathematics**
**Example Problem: Write the Sum Using Sigma Notation**
Given the arithmetic series:
\[ 3 + 4 + 5 + 6 + \ldots + 91 \]
We need to express this series using sigma notation.
### Solution:
To express this sum using sigma notation, follow these steps:
1. **Identify the First Term and the Pattern:**
- The first term of the series is 3.
- The series increases by 1 at each step (i.e., it is an arithmetic series).
2. **Determine the General Term:**
- The \(n\)-th term of this sequence can be written as \( k + 1 \), where \(k\) starts at 2 (since \(2 + 1 = 3\)) and increases by 1 each time.
3. **Determine the Range of \(k\):**
- The last term 91 can be rewritten as \( k + 1 = 91 \Rightarrow k = 90 \).
- The starting value of \(k\) is 2, as determined from the sequence.
4. **Sigma Notation:**
\[
\sum_{k=2}^{90} (k + 1)
\]
**Correction in the Image Example:**
In the image, the sigma notation for \( \sum_{k=3}^{91} (k + 1) \) is incorrect because it changes the sequence. The correct sigma notation should start the index \(k\) from 2, given that the term is \( k + 1 \).
### Conclusion:
Using sigma notation helps to represent long sums compactly, which is highly useful for computations and mathematical analysis.
(Note: The image shows an incorrect representation for this sequence with a representation indicating a start at \(k = 3\) with term \(k + 1\), which would inaccurately represent the given arithmetic sequence starting from 3.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0313a3d2-f852-4af8-9bdc-dcc8f1b4e305%2F865c3517-bf6f-48bd-9f88-fbefbc4d6dd3%2Fb3ohufh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Sigma Notation in Mathematics**
**Example Problem: Write the Sum Using Sigma Notation**
Given the arithmetic series:
\[ 3 + 4 + 5 + 6 + \ldots + 91 \]
We need to express this series using sigma notation.
### Solution:
To express this sum using sigma notation, follow these steps:
1. **Identify the First Term and the Pattern:**
- The first term of the series is 3.
- The series increases by 1 at each step (i.e., it is an arithmetic series).
2. **Determine the General Term:**
- The \(n\)-th term of this sequence can be written as \( k + 1 \), where \(k\) starts at 2 (since \(2 + 1 = 3\)) and increases by 1 each time.
3. **Determine the Range of \(k\):**
- The last term 91 can be rewritten as \( k + 1 = 91 \Rightarrow k = 90 \).
- The starting value of \(k\) is 2, as determined from the sequence.
4. **Sigma Notation:**
\[
\sum_{k=2}^{90} (k + 1)
\]
**Correction in the Image Example:**
In the image, the sigma notation for \( \sum_{k=3}^{91} (k + 1) \) is incorrect because it changes the sequence. The correct sigma notation should start the index \(k\) from 2, given that the term is \( k + 1 \).
### Conclusion:
Using sigma notation helps to represent long sums compactly, which is highly useful for computations and mathematical analysis.
(Note: The image shows an incorrect representation for this sequence with a representation indicating a start at \(k = 3\) with term \(k + 1\), which would inaccurately represent the given arithmetic sequence starting from 3.)
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