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,\[\int_{x=0}^{+\infty} x^{125} e^{-x} \, dx = \boxed{125!}.\]#### SummaryUnderstanding 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.

Name: Find #2 ### Understanding Factorials and Euler's Number (e)#### 1. Fac
Uploaded: 2024-03-20T07:06:41.000Z
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Description: 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