Indefinite sum

In mathematics the indefinite sum operator (also known as the antidifference operator), denoted by ∑ x {displaystyle sum _{x}} or Δ − 1 {displaystyle Delta ^{-1}} , is the linear operator, inverse of the forward difference operator Δ {displaystyle Delta } . It relates to the forward difference operator as the indefinite integral relates to the derivative. ThusIndefinite sums can be used to calculate definite sums with the formula:provided that the right-hand side of the equation converges.Often the constant C in indefinite sum is fixed from the following condition.Indefinite summation by parts:If T {displaystyle T}   is a period of function f ( x ) {displaystyle f(x)}   thenSome authors use the phrase 'indefinite sum' to describe a sum in which the numerical value of the upper limit is not given:This is a list of indefinite sums of various functions. Not every function has an indefinite sum that can be expressed in terms of elementary functions.