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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Question
Determine a region whose area is equal to the given limit. Do not evaluate the limit.
![The image contains the following mathematical expression:
\[
\lim_{x \to \infty} \sum_{i=1}^{n} \frac{2}{n} \cdot \frac{1}{1 + \left(\frac{2i}{n}\right)}
\]
This expression involves:
1. **Limit:** As \( x \) approaches infinity, indicating a process of finding the behavior of the expression for very large values of \( x \).
2. **Summation:** The sum is taken from \( i = 1 \) to \( n \), where each term of the sum is \(\frac{2}{n} \times \frac{1}{1 + \left(\frac{2i}{n}\right)}\).
3. **Fractional Term in the Summation:**
- The term \(\frac{2}{n}\) acts as a constant multiplied by each fraction within the summation.
- The denominator \(\left(1 + \frac{2i}{n}\right)\) represents a linear transformation inside the fraction.
The expression is a representation of a Riemann sum, which approximates an integral as \( n \) approaches infinity. It breaks the area under a curve into \( n \) rectangles and sums this area as the width of the rectangles shrinks.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee25dce1-d28a-466f-96eb-a0601f68a775%2Fc1811cf1-28af-4465-8065-340e36bb6b87%2Fkjusix9_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains the following mathematical expression:
\[
\lim_{x \to \infty} \sum_{i=1}^{n} \frac{2}{n} \cdot \frac{1}{1 + \left(\frac{2i}{n}\right)}
\]
This expression involves:
1. **Limit:** As \( x \) approaches infinity, indicating a process of finding the behavior of the expression for very large values of \( x \).
2. **Summation:** The sum is taken from \( i = 1 \) to \( n \), where each term of the sum is \(\frac{2}{n} \times \frac{1}{1 + \left(\frac{2i}{n}\right)}\).
3. **Fractional Term in the Summation:**
- The term \(\frac{2}{n}\) acts as a constant multiplied by each fraction within the summation.
- The denominator \(\left(1 + \frac{2i}{n}\right)\) represents a linear transformation inside the fraction.
The expression is a representation of a Riemann sum, which approximates an integral as \( n \) approaches infinity. It breaks the area under a curve into \( n \) rectangles and sums this area as the width of the rectangles shrinks.
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