the above equation is the following: Using binomial theorem, this may be rewritten as: The above formula is more commonly used for inverting of the difference operator , defined by: where is a function defined on the nonnegative integers. Thus, given such a function , the problem is to compute the antidifference of , a function such that . That is, This function is defined up to the addition of a constant, and may be chosen as There is not always a closed-form expression for such a summation, but Faulhaber's formula provides a closed form in the case where and, by linearity, for every polynomial function of . Approximation by definite integrals Many such approximations can be obtained by the following connection between sums and integrals, which holds for any increasing function f: and for any decreasing function f: For more general approximations, see the Euler–Maclaurin formula. For summations in which the summand is given (or can be interpolated) by an integrable function of the index, the summation can be interpreted as a Riemann sum occurring in the definition of the corresponding definite integral. One can therefore expect that for instance since the right-hand side is by definition the limit for of the left-hand side. However, for a given summation n is fixed, and little can be said about the error in the above approximation without additional assumptions about f: it is clear that for wildly oscillating functions the Riemann sum can be arbitrarily far from the Riemann integral. Identities The formulae below involve finite sums; for infinite summations or finite summations of expressions involving trigonometric functions or other transcendental functions, see list of mathematical series. General identities (distributivity) (commutativity and associativity) (index shift) for a bijection from a finite set onto a set (index change); this generalizes the preceding formula. (splitting a sum, using associativity) (a variant of the preceding formula) (the sum from the first term up to the last is equal to the sum from the last down to the first) (a particular case of the formula above) (commutativity and associativity, again) (another application of commutativity and associativity) (splitting a sum into its odd and even parts, for even indexes) (splitting a sum into its odd and even parts, for odd indexes) (distributivity) (distributivity allows factorization) (the logarithm of a product is the sum of the logarithms of the factors) (the exponential of a sum is the