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  2. Empty sum - Wikipedia

    en.wikipedia.org/wiki/Empty_sum

    In mathematics, an empty sum, or nullary sum, [1] is a summation where the number of terms is zero. The natural way to extend non-empty sums [ 2 ] is to let the empty sum be the additive identity . Let a 1 {\displaystyle a_{1}} , a 2 {\displaystyle a_{2}} , a 3 {\displaystyle a_{3}} , ... be a sequence of numbers, and let

  3. Summation - Wikipedia

    en.wikipedia.org/wiki/Summation

    Summation of a sequence of only one summand results in the summand itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0. Very often, the elements of a sequence are defined, through a regular pattern, as a function of their place in the sequence. For simple patterns, summation of long sequences may be ...

  4. List of sums of reciprocals - Wikipedia

    en.wikipedia.org/wiki/List_of_sums_of_reciprocals

    The sum of the reciprocals of the numbers in any sum-free sequence is less than 2.8570 . The sum of the reciprocals of the heptagonal numbers converges to a known value that is not only irrational but also transcendental, and for which there exists a complicated formula.

  5. Composition (combinatorics) - Wikipedia

    en.wikipedia.org/wiki/Composition_(combinatorics)

    Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same integer partition of that number. Every integer has finitely many distinct compositions. Negative numbers do not have any compositions, but 0 has one composition, the empty sequence.

  6. Summation by parts - Wikipedia

    en.wikipedia.org/wiki/Summation_by_parts

    It may be used to prove Nicomachus's theorem that the sum of the first cubes equals the square of the sum of the first positive integers. [2] Summation by parts is frequently used to prove Abel's theorem and Dirichlet's test.

  7. Iterated binary operation - Wikipedia

    en.wikipedia.org/wiki/Iterated_binary_operation

    If f has a unique left identity e, the definition of F l can be modified to operate on empty sequences by defining the value of F l on an empty sequence to be e (the previous base case on sequences of length 1 becomes redundant). Similarly, F r can be modified to operate on empty sequences if f has a unique right identity.

  8. Arithmetic progression - Wikipedia

    en.wikipedia.org/wiki/Arithmetic_progression

    For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2. If the initial term of an arithmetic progression is a 1 {\displaystyle a_{1}} and the common difference of successive members is d {\displaystyle d} , then the n {\displaystyle n} -th term of the sequence ( a n {\displaystyle a_{n ...

  9. Integer partition - Wikipedia

    en.wikipedia.org/wiki/Integer_partition

    In number theory and combinatorics, a partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.)