Can we define factorial of a negative integer ?

Answer

To determine whether we can define the factorial of a negative integer, let's analyze the concept of factorial step by step.

### Step 1: Understanding Factorial
The factorial of a non-negative integer \( n \) is defined as the product of all positive integers from 1 to \( n \). It is denoted as \( n! \) and is calculated as:
\[ n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1 \]
For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

### Step 2: Factorial of Zero
The factorial of zero, denoted as \( 0! \), is defined to be 1. This is a special case that helps in various mathematical formulations, particularly in combinatorics.

### Step 3: Factorial of Negative Integers
Now, let's consider negative integers. For a negative integer \( -n \) (where \( n \) is a positive integer), there are no positive integers to multiply. Thus, if we try to apply the factorial definition:
\[ (-n)! = (-n) \times (-n-1) \times (-n-2) \times \ldots \]
This sequence does not converge to a finite product, and therefore, we cannot define the factorial for negative integers.

### Step 4: Conclusion
From the above analysis, we conclude that the factorial of a negative integer is not defined. The factorial function is only defined for non-negative integers (0 and positive integers).

### Final Answer
No, the factorial of a negative integer is not defined.

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