section_path: Calculus and partial summation as an operation on sequences
section_name: Calculus and partial summation as an operation on sequences
paragraph_text: Calculus and partial summation as an operation on sequences
  
  Partial summation takes as input a sequence, (a), and gives as output another sequence, (S). It is thus a unary operation on sequences. Further, this function is linear, and thus is a linear operator on the vector space of sequences, denoted Σ. The inverse operator is the finite difference operator, denoted Δ. These behave as discrete analogues of integration and differentiation, only for series (functions of a natural number) instead of functions of a real variable. For example, the sequence (1, 1, 1,...) has series (1, 2, 3, 4,...) as its partial summation, which is analogous to the fact that ∫ 0 x 1 d t = x. {\textstyle \int _{0}^{x}1\,dt=x.}
  
  In computer science, it is known as prefix sum.