sequence is defined in S, then an infinite iteration on a countable sequence in S is defined exactly when the corresponding sequence of finite iterations converges. Thus, e.g., if a0, a1, a2, a3, … is an infinite sequence of real numbers, then the infinite product is defined, and equal to if and only if that limit exists. Non-associative binary operation The general, non-associative binary operation is given by a magma. The act of iterating on a non-associative binary operation may be represented as a binary tree. Notation Iterated binary operations are used to represent an operation that will be repeated over a set subject to some constraints. Typically the lower bound of a restriction is written under the symbol, and the upper bound over the symbol, though they may also be written as superscripts and subscripts in compact notation. Interpolation is performed over positive integers from the lower to upper bound, to produce the set which will be substituted into the index (below denoted as i) for the repeated operations. Common notations include the big Sigma (repeated sum) and big Pi (repeated product) notations. It is possible to specify set membership or other logical constraints in place of explicit indices, in order to implicitly specify which elements of a set shall be used: Multiple conditions may be written either joined with a logical and or separately: Less commonly, any binary operator such as exclusive or or set union may also be used. For example, if S is a set of logical propositions: which is true iff all of the elements of S are true. See also Continued fraction Fold (higher-order function) Infinite product Infinite series References External links Bulk action Parallel prefix operation Nuprl iterated binary operations Binary operations