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Exponential functions with bases 2 and 1/2. The exponential function is a mathematical function denoted by () = or (where the argument x is written as an exponent).Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras.
For example, 3 5 = 3 · 3 · 3 · 3 · 3 = 243. The base 3 appears 5 times in the multiplication, because the exponent is 5. Here, 243 is the 5th power of 3, or 3 raised to the 5th power. The word "raised" is usually omitted, and sometimes "power" as well, so 3 5 can be simply read "3 to the 5th", or "3 to
For each integer n > 2, the function n x is defined and increasing for x ≥ 1, and n 1 = 1, so that the n th super-root of x, , exists for x ≥ 1. One of the simpler and faster formulas for a third-degree super-root is the recursive formula, if: x x x = a , and next x ( n + 1) = exp (W (W ( x ( n ) ln ( a )))) , for example x (0) = 1 .
Comparison of Stirling's approximation with the factorial. In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of . It is named after James Stirling, though a related but less precise result was first stated ...
Fermat sent the letters in which he mentioned the case in which n = 3 in 1636, 1640 and 1657. [31] Euler sent a letter to Goldbach on 4 August 1753 in which claimed to have a proof of the case in which n = 3. [32] Euler had a complete and pure elementary proof in 1760, but the result was not published. [33] Later, Euler's proof for n = 3 was ...
Euler derived the formula as connecting a finite sum of products with a finite continued fraction. (+ (+ (+))) = + + + + = + + + +The identity is easily established by induction on n, and is therefore applicable in the limit: if the expression on the left is extended to represent a convergent infinite series, the expression on the right can also be extended to represent a convergent infinite ...
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, it is possible to expand the polynomial (x + y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending ...
A formal power series is a special kind of formal series, of the form. where the called coefficients, are numbers or, more generally, elements of some ring, and the are formal powers of the symbol that is called an indeterminate or, commonly, a variable. Hence, power series can be viewed as a generalization of polynomials where the number of ...