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
![**14. Sum to integral:**
Evaluate the following limit by identifying the integral that it represents:
\[ \lim_{n \to \infty} \sum_{k=1}^{n} \left( \left( \frac{4k}{n} \right)^5 + 1 \right) \frac{4}{n}. \]
---
**Explanation:**
Given the problem, we notice that the expression \(\frac{4k}{n}\) can be identified as the partition points of the interval \([0, 4]\) as \(n \to \infty\). The expression within the summation therefore represents a Riemann sum.
To convert this into an integral, observe the following transformation:
1. Identify the function being summed. Let's denote it as \(f(x)\), where \(x = \frac{4k}{n}\). So, \(f(x) = x^5 + 1\).
2. The term \(\frac{4}{n}\) represents the width \(\Delta x\) of each subinterval when the interval \([0, 4]\) is partitioned uniformly into \(n\) subintervals.
Therefore, as \(n \to \infty\),
\[ \sum_{k=1}^{n} \left( \left( \frac{4k}{n} \right)^5 + 1 \right) \frac{4}{n} \]
becomes
\[ \int_{0}^{4} (x^5 + 1) \, dx. \]
So, we evaluate the integral:
\[ \int_{0}^{4} (x^5 + 1) \, dx. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd4248fea-c75e-4e96-a274-f1cd3c8da07a%2F27d8eb4b-6d1e-4b10-9732-bb2f0f2429fa%2Fftlaano_processed.png&w=3840&q=75)
Transcribed Image Text:**14. Sum to integral:**
Evaluate the following limit by identifying the integral that it represents:
\[ \lim_{n \to \infty} \sum_{k=1}^{n} \left( \left( \frac{4k}{n} \right)^5 + 1 \right) \frac{4}{n}. \]
---
**Explanation:**
Given the problem, we notice that the expression \(\frac{4k}{n}\) can be identified as the partition points of the interval \([0, 4]\) as \(n \to \infty\). The expression within the summation therefore represents a Riemann sum.
To convert this into an integral, observe the following transformation:
1. Identify the function being summed. Let's denote it as \(f(x)\), where \(x = \frac{4k}{n}\). So, \(f(x) = x^5 + 1\).
2. The term \(\frac{4}{n}\) represents the width \(\Delta x\) of each subinterval when the interval \([0, 4]\) is partitioned uniformly into \(n\) subintervals.
Therefore, as \(n \to \infty\),
\[ \sum_{k=1}^{n} \left( \left( \frac{4k}{n} \right)^5 + 1 \right) \frac{4}{n} \]
becomes
\[ \int_{0}^{4} (x^5 + 1) \, dx. \]
So, we evaluate the integral:
\[ \int_{0}^{4} (x^5 + 1) \, dx. \]
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