Summation

Capital-sigma notation Mathematical notation uses a symbol that compactly represents summation of many similar terms: the summation symbol, \sum, an enlarged form of the upright capital Greek letter sigma. This is defined as \sum_{i \mathop =m}^n a_i = a_m + a_{m+1} + a_{m+2} + \cdots + a_{n-1} + a_n where is the "index of summation" or "dummy variable", is an indexed variable representing each term of the sum; is the "lower bound of summation", and is the "upper bound of summation". The "" under the summation symbol means that the index starts out equal to . The index, , is incremented by one for each successive term, stopping when . This is read as "sum of , from to ". However, some notations may include the index at the upper bound of summation, or omit the index at the lower bound as in \sum_{i=m} ^{i=n} a_i or \sum_m ^n a_i , respectively. There are sigma notation variants where the range of bounds is omitted, which denotes the dummy variable only, like \sum_i a_i . Here is an example showing the summation of squares: \sum_{i = 3}^6 i^2 = 3^2+4^2+5^2+6^2 = 86. In general, while any variable can be used as the index of summation (provided that no ambiguity is incurred), some of the most common ones include letters such as i, j, k, and n; the latter is also often used for the upper bound of a summation. Generalizations of this notation are often used, in which an arbitrary logical condition is supplied, and the sum is intended to be taken over all values satisfying the condition. For example, \sum_{0 \le k is an alternative notation for \sum_{k = 0}^{99} f(k), the sum of f(k) over all (integers) k in the specified range. 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 \sum (2wx + w^2) = x^2. The usage of sigma notation was later attested by mathematicians such as Lagrange, who denoted \sum and \sum ^n in 1772. Fourier and C. G. J. Jacobi also denoted the sigma notation in 1829, but Fourier included lower and upper bounds as in \sum_{i=1}^{\infty}e^{-i^2t} \ldots. Other than sigma notation, the capital letter S is attested as a summation symbol for series in 1823, which was apparently widespread. == Formal definition ==

Summation may be defined recursively as an operator, taking in a function and two natural numbers, as follows: :\sum : (\R^\R \times \N \times \N) \mapsto \R, such that: : :\sum_{i=a}^b g(i)=0, for b; : :\sum_{i=a}^b g(i)=g(b)+\sum_{i=a}^{b-1} g(i), for b \geqslant a. ==Measure theory notation==

In the notation of measure and integration theory, a sum can be expressed as a definite integral, :\sum_{k \mathop =a}^b f(k) = \int_{[a,b]} f\,d\mu where [a, b] is the subset of the integers from a to b, and where \mu is the counting measure over the integers. ==Calculus of finite differences==

Given a function that is defined over the integers in the interval , the following equation holds: :f(n)-f(m)= \sum_{i=m}^{n-1} (f(i+1)-f(i)). This is known as a telescoping series and is the analogue of the fundamental theorem of calculus in calculus of finite differences, which states that: :f(n)-f(m)=\int_m^n f'(x)\,dx, where :f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h} is the derivative of . An example of application of the above equation is the following: :n^k=\sum_{i=0}^{n-1} \left((i+1)^k-i^k\right). Using binomial theorem, this may be rewritten as: :n^k=\sum_{i=0}^{n-1} \biggl(\sum_{j=0}^{k-1} \binom{k}{j} i^j\biggr). The above formula is more commonly used for inverting of the difference operator \Delta, defined by: :\Delta(f)(n)=f(n+1)-f(n), where is a function defined on the nonnegative integers. Thus, given such a function , the problem is to compute the antidifference of , a function F=\Delta^{-1}f such that \Delta F=f. That is, F(n+1)-F(n)=f(n). This function is defined up to the addition of a constant, and may be chosen as :F(n)=\sum_{i=0}^{n-1} f(i). There is not always a closed-form expression for such a summation, but Faulhaber's formula provides a closed form in the case where f(n)=n^k 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: :\int_{s=a-1}^{b} f(s)\ ds \le \sum_{i=a}^{b} f(i) \le \int_{s=a}^{b+1} f(s)\ ds. and for any decreasing function f: :\int_{s=a}^{b+1} f(s)\ ds \le \sum_{i=a}^{b} f(i) \le \int_{s=a-1}^{b} f(s)\ ds. 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 :\frac{b-a}{n}\sum_{i=0}^{n-1} f\left(a+i\frac{b-a}n\right) \approx \int_a^b f(x)\ dx, since the right-hand side is by definition the limit for n\to\infty 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 : \sum_{n=s}^t C\cdot f(n) = C\cdot \sum_{n=s}^t f(n) \quad (distributivity) : \sum_{n=s}^t f(n) \pm \sum_{n=s}^{t} g(n) = \sum_{n=s}^t \left(f(n) \pm g(n)\right)\quad (commutativity and associativity) : \sum_{n=s}^t f(n) = \sum_{n=s+p}^{t+p} f(n-p)\quad (index shift) : \sum_{n\in B} f(n) = \sum_{m\in A} f(\sigma(m)), \quad for a bijection from a finite set onto a set (index change); this generalizes the preceding formula. : \sum_{n=s}^t f(n) =\sum_{n=s}^j f(n) + \sum_{n=j+1}^t f(n)\quad (splitting a sum, using associativity) : \sum_{n=a}^{b}f(n)=\sum_{n=0}^{b}f(n)-\sum_{n=0}^{a-1}f(n)\quad (a variant of the preceding formula) : \sum_{n=s}^t f(n) = \sum_{n=0}^{t-s} f(t-n)\quad (the sum from the first term up to the last is equal to the sum from the last down to the first) : \sum_{n=0}^t f(n) = \sum_{n=0}^{t} f(t-n)\quad (a particular case of the formula above) : \sum_{i=k_0}^{k_1}\sum_{j=l_0}^{l_1} a_{i,j} = \sum_{j=l_0}^{l_1}\sum_{i=k_0}^{k_1} a_{i,j}\quad (commutativity and associativity, again) : \sum_{k\le j \le i\le n} a_{i,j} = \sum_{i=k}^n\sum_{j=k}^i a_{i,j} = \sum_{j=k}^n\sum_{i=j}^n a_{i,j} = \sum_{j=0}^{n-k}\sum_{i=k}^{n-j} a_{i+j,i}\quad (another application of commutativity and associativity) : \sum_{n=2s}^{2t+1} f(n) = \sum_{n=s}^t f(2n) + \sum_{n=s}^t f(2n+1)\quad (splitting a sum into its odd and even parts, for even indexes) : \sum_{n=2s+1}^{2t} f(n) = \sum_{n=s+1}^t f(2n) + \sum_{n=s+1}^t f(2n-1)\quad (splitting a sum into its odd and even parts, for odd indexes) : \sum_{n=s}^t \log_b f(n) = \log_b \prod_{n=s}^t f(n)\quad (the logarithm of a product is the sum of the logarithms of the factors) : C^{\sum\limits_{n=s}^t f(n) } = \prod_{n=s}^t C^{f(n)}\quad (the exponential of a sum is the product of the exponential of the summands) : \sum^{k}_{m = 0}\sum^{m}_{n = 0}f(m,n)=\sum^{k}_{m = 0}\sum^{k}_{n = m}f(n,m),\quadfor any function f from \mathbb{Z}\times\mathbb{Z}. Powers and logarithm of arithmetic progressions : \sum_{i=1}^n c = nc\quad for every that does not depend on : \sum_{i=0}^n i = \sum_{i=1}^n i = \frac{n(n+1)}{2}\qquad (Sum of the simplest arithmetic progression, consisting of the first n natural numbers.) : \sum_{i=1}^n (2i-1) = n^2\qquad (Sum of first odd natural numbers) : \sum_{i=0}^{n} 2i = n(n+1)\qquad (Sum of first even natural numbers) : \sum_{i=1}^{n} \log i = \log (n!)\qquad (A sum of logarithms is the logarithm of the product) : \sum_{i=0}^n i^2 = \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} = \frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6}\qquad (Sum of the first squares, see square pyramidal number.) : \sum_{i=0}^n i^3 = \biggl(\sum_{i=0}^n i \biggr)^2 = \left(\frac{n(n+1)}{2}\right)^2 = \frac{n^4}{4} + \frac{n^3}{2} + \frac{n^2}{4}\qquad (Nicomachus's theorem) More generally, one has Faulhaber's formula for p>1 : \sum_{k=1}^n k^{p} = \frac{n^{p+1}}{p+1} + \frac{1}{2}n^p + \sum_{k=2}^p \binom p k \frac{B_k}{p-k+1}\,n^{p-k+1}, where B_k denotes a Bernoulli number, and \binom p k is a binomial coefficient. Summation index in exponents In the following summations, is assumed to be different from 1. : \sum_{i=0}^{n-1} a^i = \frac{1-a^n}{1-a} (sum of a geometric progression) : \sum_{i=0}^{n-1} \frac{1}{2^i} = 2-\frac{1}{2^{n-1}} (special case for ) : \sum_{i=0}^{n-1} i a^i =\frac{a-na^n+(n-1)a^{n+1}}{(1-a)^2} ( times the derivative with respect to of the geometric progression) : \begin {align} \sum_{i= 0}^{n-1} \left(b + i d\right) a^i &= b \sum_{i= 0}^{n-1} a^i + d \sum_{i= 0}^{n-1} i a^i\\ & = b \left(\frac{1-a^n}{1-a}\right) + d \left(\frac{a-na^n+(n-1)a^{n+1}}{(1-a)^2}\right)\\ & = \frac{b(1-a^n) - (n - 1)d a^n}{1 - a}+\frac{da(1 - a^{n - 1})}{(1 - a)^2} \end {align} :::(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 : \sum_{i=0}^n {n \choose i}a^{n-i} b^i=(a + b)^n, the binomial theorem : \sum_{i=0}^n {n \choose i} = 2^n, the special case where : \sum_{i=0}^n {n \choose i}p^i (1-p)^{n-i}=1, the special case where , which, for 0 \le p \le 1, expresses the sum of the binomial distribution : \sum_{i=0}^{n} i{n \choose i} = n(2^{n-1}), the value at of the derivative with respect to of the binomial theorem : \sum_{i=0}^n \frac{n \choose i}{i+1} = \frac{2^{n+1}-1}{n+1}, the value at of the antiderivative with respect to of the binomial theorem Involving permutation numbers In the following summations, {}_{n}P_{k} is the number of -permutations of. : \sum_{i=0}^{n} {}_{i}P_{k}{n \choose i} = {}_{n}P_{k}(2^{n-k}) : \sum_{i=1}^n {}_{i+k}P_{k+1} = \sum_{i=1}^n \prod_{j=0}^k (i+j) = \frac{(n+k+1)!}{(n-1)!(k+2)} : \sum_{i=0}^{n} i!\cdot{n \choose i} = \sum_{i=0}^{n} {}_{n}P_{i} = \lfloor n! \cdot e \rfloor, \quad n \in \mathbb{Z}^+, where and \lfloor x\rfloor denotes the floor function. Others : \sum_{k=0}^{m} \binom{n+k}{n} = \binom{n+m+1}{n+1} : \sum_{i=k}^{n} {i \choose k} = {n+1 \choose k+1} : \sum_{i=0}^n i\cdot i! = (n+1)! - 1 : \sum_{i=0}^n {m+i-1 \choose i} = {m+n \choose n} :\sum_{i=0}^n {n \choose i}^2 = {2n \choose n} :\sum_{i=0}^n \frac{1}{i!} = \frac{\lfloor n!\; e \rfloor}{n!} Harmonic numbers : \sum_{i=1}^n \frac{1}{i} = H_n\quad (the th harmonic number) : \sum_{i=1}^n \frac{1}{i^k} = H^{(k)}_n\quad (a generalized harmonic number) ==Growth rates==

The following are useful approximations (using theta notation): : \sum_{i=1}^n i^c \in \Theta(n^{c+1}) for real c greater than −1 : : \sum_{i=1}^n \frac{1}{i} \in \Theta(\log_e n) (See Harmonic number) : : \sum_{i=1}^n c^i \in \Theta(c^n) for real c greater than 1 : : \sum_{i=1}^n \log(i)^c \in \Theta(n \cdot \log(n)^{c}) for non-negative real c : : \sum_{i=1}^n \log(i)^c \cdot i^d \in \Theta(n^{d+1} \cdot \log(n)^{c}) for non-negative real c, d : : \sum_{i=1}^n \log(i)^c \cdot i^d \cdot b^i \in \Theta (n^d \cdot \log(n)^c \cdot b^n) for non-negative real b > 1, c, d ==See also==