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The exponential factorial is a positive integer n raised to the power of n − 1, ... The first few exponential factorials are 1, 2, 9, 262144, ...
Calculators may associate exponents to the left or to the right. For example, the expression a ^ b ^ c is interpreted as a ( b c ) on the TI-92 and the TI-30XS MultiView in "Mathprint mode", whereas it is interpreted as ( a b ) c on the TI-30XII and the TI-30XS MultiView in "Classic mode".
The exponential factorial is defined recursively as =, =. For example, the exponential factorial of 4 is 4 3 2 1 = 262144. {\displaystyle 4^{3^{2^{1}}}=262144.} These numbers grow much more quickly than regular factorials.
The exponential function is the sum of the power series [2] [3] = + +! +! + = =!, where ! is the factorial of n (the product of the n first positive integers). This series is absolutely convergent for every x {\displaystyle x} per the ratio test .
In this article, the symbol () is used to represent the falling factorial, and the symbol () is used for the rising factorial. These conventions are used in combinatorics , [ 4 ] although Knuth 's underline and overline notations x n _ {\displaystyle x^{\underline {n}}} and x n ¯ {\displaystyle x^{\overline {n}}} are increasingly popular.
A simple arithmetic calculator was first included with Windows 1.0. [5]In Windows 3.0, a scientific mode was added, which included exponents and roots, logarithms, factorial-based functions, trigonometry (supports radian, degree and gradians angles), base conversions (2, 8, 10, 16), logic operations, statistical functions such as single variable statistics and linear regression.
Consider the factorial function F(n) recursively defined by F(n) = 1, if n = 0; else n × F(n − 1). In the lambda expression which is to represent this function, a parameter (typically the first one) will be assumed to receive the lambda expression itself as its value, so that calling it – applying it to an argument – will amount to ...
In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...