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Stirling's approximation provides an accurate approximation to the factorial of large numbers, showing that it grows more quickly than exponential growth. Legendre's formula describes the exponents of the prime numbers in a prime factorization of the factorials, and can be used to count the trailing zeros of the factorials.
The exponential factorials grow much more quickly than regular factorials or even hyperfactorials. The number of digits in the exponential factorial of 6 is approximately 5 × 10 183 230. The sum of the reciprocals of the exponential factorials from 1 onwards is the following transcendental number:
For other finite sums of subsets of the reciprocals of Fibonacci numbers, see here. An exponential factorial is an operation recursively defined as =, = . For example, = where the exponents are evaluated from the top down. The sum of the reciprocals of the exponential factorials from 1 onward is approximately 1.6111 and is transcendental.
Fractions. Casio continued to offer their unique rational number arithmetic first introduced in the AL-10, along with their novel use of an L-shaped character to display "over". Due to limitations both in working with rationals and the display, only 3 digit numerators or denominators are allowed, otherwise the value reverts to floating point .
Here is a sample program that computes the factorial of an integer number from 2 to 69 (ignoring the calculator's built-in factorial/gamma function). There are two versions of the example: one for algebraic mode and one for RPN mode. The RPN version is significantly shorter. Algebraic version:
While the first interpretation may be expected by some users due to the nature of implied multiplication, [38] the latter is more in line with the rule that multiplication and division are of equal precedence. [3] When the user is unsure how a calculator will interpret an expression, parentheses can be used to remove the ambiguity. [3]
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
The falling factorial can be extended to real values of using the gamma function provided and + are real numbers that are not negative integers: = (+) (+) , and so can the rising factorial: = (+) . Calculus