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The Fermat–Catalan conjecture generalizes Fermat's last theorem with the ideas of the Catalan conjecture. [ 167 ] [ 168 ] The conjecture states that the generalized Fermat equation has only finitely many solutions ( a , b , c , m , n , k ) with distinct triplets of values ( a m , b n , c k ), where a , b , c are positive coprime integers and ...
The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US $1 million prize for the first correct solution to each problem.
The series' terms (1, −2, 3, −4, ...) do not approach 0; therefore 1 − 2 + 3 − 4 + ... diverges by the term test.Divergence can also be shown directly from ...
Those methods work on oscillating divergent series, but they cannot produce a finite answer for a series that diverges to +∞. [6] Most of the more elementary definitions of the sum of a divergent series are stable and linear, and any method that is both stable and linear cannot sum 1 + 2 + 3 + ⋯ to a finite value (see § Heuristics below).
The term is often used exclusively to refer to linear time-invariant (LTI) systems. Most real systems have non-linear input-output characteristics, but many systems operated within nominal parameters (not over-driven) have behavior close enough to linear that LTI system theory is an acceptable representation of their input-output behavior.
The sum of all such rectangles gives an approximation of the area between the axis and the curve, which is an approximation of the total distance traveled. A smaller value for Δx will give more rectangles and in most cases a better approximation, but for an exact answer, we need to take a limit as Δx approaches zero. [46]: 512–522
y = x 3 for values of 1 ≤ x ≤ 25.. In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example 2 3 = 8 or (x + 1) 3.
It is unknown whether these constants are transcendental in general, but Γ( 1 / 3 ) and Γ( 1 / 4 ) were shown to be transcendental by G. V. Chudnovsky. Γ( 1 / 4 ) / 4 √ π has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that Γ( 1 / 4 ), π, and e π are algebraically independent.