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In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
In number theory, Ramanujan's sum, usually denoted c q (n), is a function of two positive integer variables q and n defined by the formula c q ( n ) = ∑ 1 ≤ a ≤ q ( a , q ) = 1 e 2 π i a q n , {\displaystyle c_{q}(n)=\sum _{1\leq a\leq q \atop (a,q)=1}e^{2\pi i{\tfrac {a}{q}}n},}
A summation method that is linear and stable cannot sum the series 1 + 2 + 3 + ⋯ to any finite value. (Stable means that adding a term at the beginning of the series increases the sum by the value of the added term.) This can be seen as follows. If + + + =, then adding 0 to both sides gives
Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series.Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined.
The sum of the series is approximately equal to 1.644934. [3] The Basel problem asks for the exact sum of this series (in closed form ), as well as a proof that this sum is correct. Euler found the exact sum to be π 2 / 6 {\displaystyle \pi ^{2}/6} and announced this discovery in 1735.
2.3 Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship. ... 7.2 Sum of reciprocal of factorials. 7.3 Trigonometry and ...
Euler's sum of powers conjecture § k = 3, relating to cubes that can be written as a sum of three positive cubes; Plato's number, an ancient text possibly discussing the equation 3 3 + 4 3 + 5 3 = 6 3; Taxicab number, the smallest integer that can be expressed as a sum of two positive integer cubes in n distinct ways
A summation-by-parts (SBP) finite difference operator conventionally consists of a centered difference interior scheme and specific boundary stencils that mimics behaviors of the corresponding integration-by-parts formulation. [3] [4] The boundary conditions are usually imposed by the Simultaneous-Approximation-Term (SAT) technique. [5]