product of the exponential of the summands) Powers and logarithm of arithmetic progressions for every that does not depend on (Sum of the simplest arithmetic progression, consisting of the first n natural numbers.) (Sum of first odd natural numbers) (Sum of first even natural numbers) (A sum of logarithms is the logarithm of the product) (Sum of the first squares, see square pyramidal number.) (Nicomachus's theorem) More generally, one has Faulhaber's formula for where denotes a Bernoulli number, and is a binomial coefficient. Summation index in exponents In the following summations, is assumed to be different from 1. (sum of a geometric progression) (special case for ) ( times the derivative with respect to of the geometric progression) (sum of an arithmetico–geometric sequence) Binomial coefficients and factorials There exist very many summation identities involving binomial coefficients (a whole chapter of Concrete Mathematics is devoted to just the basic techniques). Some of the most basic ones are the following. Involving the binomial theorem the binomial theorem the special case where , the special case where , which, for expresses the sum of the binomial distribution the value at of the derivative with respect to of the binomial theorem the value at of the antiderivative with respect to of the binomial theorem Involving permutation numbers In the following summations, is the number of -permutations of . , where and denotes the floor function. Others Harmonic numbers (the th harmonic number) (a generalized harmonic number) Growth rates The following are useful approximations (using theta notation): for real c greater than −1 (See Harmonic number) for real c greater than 1 for non-negative real c for non-negative real c, d for non-negative real b > 1, c, d History In 1675, Gottfried Wilhelm Leibniz, in a letter to Henry Oldenburg, suggests the symbol ∫ to mark the sum of differentials (Latin: calculus summatorius), hence the S-shape. The renaming of this symbol to integral arose later in exchanges with Johann Bernoulli. In 1755, the summation symbol Σ is attested in Leonhard Euler's Institutiones calculi differentialis. Euler uses the symbol in expressions like: In 1772, usage of Σ and Σn is attested by Lagrange. In 1823, the capital letter S is attested as a summation symbol for series. This usage was apparently widespread. In 1829, the summation symbol Σ is attested by Fourier and C. G. J. Jacobi. Fourier's use includes lower and upper bounds, for