Formulas and Tables

Elementary Algebra

Factorial

Natural numbers: \(n\), \(k\)
Real numbers: \(x\)

Factorial of \(n\): \(n!\)
Gamma function: \(\Gamma \left( x \right)\)

  1. The factorial of a non-negative integer \(n\) is called the product of all positive integers less than or equal to the number \(n.\) The factorial of \(n\) is denoted as \(n!\)
    \(n! =\) \( 1 \cdot 2 \cdot 3 \ldots \left( {n – 1} \right) \cdot n\)
  2. factorial of zero by definition is equal to\(1:\) \(0! = 1\)
  3. Factorials of the numbers \(1-10\)
Factorials of the numbers 1−10
  1. Recursive formula \(\left( {n + 1} \right)! = n! \cdot \left( {n + 1} \right)\)
  2. Extension of the factorial function to non-negative real numbers
    The factorial of a non-negative real number \(x\) is expressed through the gamma function by the formula \(x! = \Gamma \left( {x + 1} \right),\) which allows to calculate the factorial of any real numbers \(x \ge 0\).
  3. Rate of increase
    The factorial function increases faster than the exponential function. The inequality \(n! \gt \exp \left( n \right)\) holds for all \(n \ge 6\). When \(n \ge 1,\) the following relation is valid: \(n \le n! \le {n^n}\).
  4. Stirling formula
    For large \(n\) the approximating factorial value can be determined by the Stirling formula:
    \(n! \approx\) \( {n^n}\sqrt {2\pi n} \,\exp \left( { – n} \right) \cdot\) \(\Big[ {1 + {\large\frac{1}{{12n}}\normalsize} + {\large\frac{1}{{288{n^2}}}\normalsize} }-\) \({ {\large\frac{{139}}{{51840{n^3}}}\normalsize} – \ldots } \Big]\).

    Taking into account only the first term in the expansion, this formula takes the form
    \(n! \approx {n^n}\sqrt {2\pi n} \,\exp \left( { – n} \right)\).

  5. Double factorial
    The double factorial is the product of all odd integers from \(1\) to some odd positive integer \(n\). The double factorial is denoted by \(n!!\)
    \(\left( {2k + 1} \right)!! =\) \( 1 \cdot 3 \cdot 5 \cdots \left( {2k – 1} \right) \cdot\) \( \left( {2k + 1} \right).\)
    Sometimes, the double factorial is considered for even integers. We can define it as
    \(\left( {2k} \right)!! =\) \( 2 \cdot 4 \cdot 6 \cdots \left( {2k – 2} \right) \cdot 2k\)
    It is implied here that \(0!! = 1.\)