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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Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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Question
Find #2
![### Understanding Factorials and Euler's Number (e)
#### 1. Factorial of Zero
**Question:** How much is \(0!\)?
**Answer:**
\[
0! = \boxed{1}
\]
(by convention).
The factorial of zero is conventionally defined as one.
#### 2. Infinite Sum Definition of \( e \)
**Question:** Give the infinite sum definition of \( e \) using factorial numbers.
**Answer:**
\[
e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \ldots
\]
(Don't forget the " + ... " at the end to indicate the continuation of the series.)
Euler's number, \(e\), can be defined as the sum of the infinite series of the reciprocals of factorials.
#### 3. Calculating a Definite Integral
**Question:** Find the integral
\[
\int_{x=0}^{+\infty} x^{125} e^{-x} \, dx.
\]
**Answer:**
This integral can be evaluated using the gamma function, \(\Gamma(n)\):
\[
\Gamma(n) = \int_{0}^{\infty} x^{n-1} e^{-x} \, dx.
\]
For \(n = 126\), we have:
\[
\Gamma(126) = \int_{0}^{\infty} x^{125} e^{-x} \, dx = 125!.
\]
Hence,
\[
\int_{x=0}^{+\infty} x^{125} e^{-x} \, dx = \boxed{125!}.
\]
#### Summary
Understanding the conventional definition of factorials and how Euler's number \(e\) can be represented as an infinite sum is crucial in many areas of mathematics. Additionally, the gamma function provides a valuable tool for evaluating complex integrals involving exponential functions and powers.
---
This transcription provides a detailed explanation for the factorial of zero, the infinite sum representation of Euler's number \(e\), and the evaluation of a specific integral using the gamma function, making it suitable for educational purposes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F239df334-5b03-4bc0-9001-d398335a6cd3%2F28612b98-8810-40bd-bf28-bad661b9e31d%2F053rkvj_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Factorials and Euler's Number (e)
#### 1. Factorial of Zero
**Question:** How much is \(0!\)?
**Answer:**
\[
0! = \boxed{1}
\]
(by convention).
The factorial of zero is conventionally defined as one.
#### 2. Infinite Sum Definition of \( e \)
**Question:** Give the infinite sum definition of \( e \) using factorial numbers.
**Answer:**
\[
e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \ldots
\]
(Don't forget the " + ... " at the end to indicate the continuation of the series.)
Euler's number, \(e\), can be defined as the sum of the infinite series of the reciprocals of factorials.
#### 3. Calculating a Definite Integral
**Question:** Find the integral
\[
\int_{x=0}^{+\infty} x^{125} e^{-x} \, dx.
\]
**Answer:**
This integral can be evaluated using the gamma function, \(\Gamma(n)\):
\[
\Gamma(n) = \int_{0}^{\infty} x^{n-1} e^{-x} \, dx.
\]
For \(n = 126\), we have:
\[
\Gamma(126) = \int_{0}^{\infty} x^{125} e^{-x} \, dx = 125!.
\]
Hence,
\[
\int_{x=0}^{+\infty} x^{125} e^{-x} \, dx = \boxed{125!}.
\]
#### Summary
Understanding the conventional definition of factorials and how Euler's number \(e\) can be represented as an infinite sum is crucial in many areas of mathematics. Additionally, the gamma function provides a valuable tool for evaluating complex integrals involving exponential functions and powers.
---
This transcription provides a detailed explanation for the factorial of zero, the infinite sum representation of Euler's number \(e\), and the evaluation of a specific integral using the gamma function, making it suitable for educational purposes.
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