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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.
t(n) = C(n + 1, 2) = n(n + 1) / 2 = 1 + 2 + ... + n for n ≥ 1, with t(0) = 0 (empty sum). A000217: Square numbers n 2: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ... n 2 = n × n: A000290: Tetrahedral numbers T(n) 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, ... T(n) is the sum of the first n triangular numbers, with T(0) = 0 (empty sum). A000292 ...
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 = (,) =,where (a, q) = 1 means that a only takes on values coprime to q.
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 nth partial sum is given by a simple formula: = = (+). This equation was known to the Pythagoreans as early as the sixth century BCE. [5] Numbers of this form are called triangular numbers, because they can be arranged as an equilateral triangle.
Pairwise summation is the default summation algorithm in NumPy [9] and the Julia technical-computing language, [10] where in both cases it was found to have comparable speed to naive summation (thanks to the use of a large base case).
The algorithm performs summation with two accumulators: sum holds the sum, and c accumulates the parts not assimilated into sum, to nudge the low-order part of sum the next time around. Thus the summation proceeds with "guard digits" in c , which is better than not having any, but is not as good as performing the calculations with double the ...
Computation of the sum 2 + 5 + 8 + 11 + 14. When the sequence is reversed and added to itself term by term, the resulting sequence has a single repeated value in it, equal to the sum of the first and last numbers (2 + 14 = 16). Thus 16 × 5 = 80 is twice the sum.