Summation

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In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.

Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article.

The summation of an explicit sequence is denoted as a succession of additions. For example, summation of Template:Math is denoted Template:Math, and results in 9, that is, Template:Math. Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one summand results in the summand itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0.

Very often, the elements of a sequence are defined, through a regular pattern, as a function of their place in the sequence. For simple patterns, summation of long sequences may be represented with most summands replaced by ellipses. For example, summation of the first 100 natural numbers may be written as Template:Math. Otherwise, summation is denoted by using Σ notation, where <math display="inline">\sum</math> is an enlarged capital Greek letter sigma. For example, the sum of the first Template:Mvar natural numbers can be denoted as

For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find closed-form expressions for the result. For example,Template:Efn

Although such formulas do not always exist, many summation formulas have been discovered—with some of the most common and elementary ones being listed in the remainder of this article.

NotationEdit

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Capital-sigma notationEdit

File:Sigma summation notation.svg

An explanation of the sigma (Σ) summation notation

Mathematical notation uses a symbol that compactly represents summation of many similar terms: the summation symbol, <math display="inline">\sum</math>, an enlarged form of the upright capital Greek letter sigma.<ref>Template:Cite book</ref> This is defined as <math display="block">\sum_{i \mathop =m}^n a_i = a_m + a_{m+1} + a_{m+2} + \cdots + a_{n-1} + a_n</math> where Template:Math is the "index of summation" or "dummy variable"Template:Sfnp, Template:Math is an indexed variable representing each term of the sum; Template:Math is the "lower bound of summation", and Template:Math is the "upper bound of summation". The "Template:Math" under the summation symbol means that the index Template:Math starts out equal to Template:Math. The index, Template:Math, is incremented by one for each successive term, stopping when Template:Math.Template:Efn This is read as "sum of Template:Math, from Template:Math to Template:Math". However, some notations may include the index at the upper bound of summation, or omit the indec at the lower bound as in <math display="inline"> \sum_{i=m} ^{i=n} a_i </math> or <math display="inline"> \sum_m ^n a_i </math>, respectively.Template:Sfnp In some cases, there are sigma notation where the range of bounds is omitted, which denotes the dummy variable only, like <math display="inline"> \sum_i a_i </math>.Template:Sfnp Here is an example showing the summation of squares: <math display="block">\sum_{i = 3}^6 i^2 = 3^2+4^2+5^2+6^2 = 86.</math> 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 <math>i</math>,Template:Efn <math>j</math>, <math>k</math>, and <math>n</math>; the latter is also often used for the upper bound of a summation.<ref name="franco"/> Alternatively, the index and bounds of summation are sometimes omitted from the definition of summation if the context is sufficiently clear. This applies particularly when the index runs from 1 to n. For example, one might write that <math display="inline">\sum a_i = \sum_{i = 1}^n a_i</math>.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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, <math display="inline">\sum_{0 \le k < 100} f(k)</math> is an alternative notation for <math display=inline>\sum_{k = 0}^{99} f(k),</math> the sum of <math>f(k)</math> over all (integers) <math>k</math> in the specified range.<ref name="franco">Template:Cite book</ref> Similarly, <math display="inline">\sum_{x \mathop \in S} f(x)</math> is the sum of <math>f(x)</math> over all elements <math>x</math> in the set <math>S</math>,<ref>Template:Cite book</ref><ref>Template:Cite book</ref> and <math display="inline">\sum_{d\,|\,n}\;\mu(d)</math> is the sum of <math>\mu(d)</math> over all positive integers <math>d</math> dividing <math>n</math>.Template:Efn

There are also ways to generalize the use of many sigma notations. For example, one writes double summation as two sigma notations with different dummy variables <math display="inline"> \sum_{i=\ell}^n \sum_{j=m}^k a_{i,j} </math>. Considering that the both sigma notation's range are the same, the double sigma notations can be wrapped into a single notation, so the double summation is rewritten as <math display="inline">\sum_{i=m}^n \sum_{j=m}^n a_{i,j} = \sum_{i,j=m}^n a_{i,j}</math>.Template:Sfnp

The term Template:Vanchor is sometimes used when discussing the summation presented above. Contrast to the infinite series, the upper bound tends to infinity <math display="inline"> \sum_{i=m}^\infty a_i </math>, which results in converge if there is a result of the sum, or diverge if otherwise. The bound in the infinite series's sigma notation can be alternatively denoted as <math display="inline"> \sum_{i \ge 0} a_i </math>.Template:Sfnp

Relatedly, the similar notation is used for the product of a sequence, where <math display="inline">\prod</math>, an enlarged form of the Greek capital letter pi, is used instead of <math display="inline">\sum</math>.Template:Sfnp

Special casesEdit

It is possible to sum fewer than 2 numbers:

If the summation has one summand <math>x</math>, then the evaluated sum is <math>x</math>.
If the summation has no summands, then the evaluated sum is zero, because zero is the identity for addition. This is known as the empty sum.

These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case. For example, if <math>n=m</math> in the definition above, then there is only one term in the sum; if <math>n=m-1</math>, then there is none.

Algebraic sumEdit

The phrase 'algebraic sum' refers to a sum of terms which may have positive or negative signs. Terms with positive signs are added, while terms with negative signs are subtracted. e.g. +1 −1

HistoryEdit

The origin of the summation notation dates back to 1675 when Gottfried Wilhelm Leibniz, in a letter to Henry Oldenburg, suggested the symbol <math display="inline"> \int </math> to mark the sum of differentials (Latin: calculus summatorius), hence the S-shape.<ref>Template:Cite book</ref><ref>Template:Cite book</ref>Template:Sfnp The renaming of this symbol to integral arose later in exchanges with Johann Bernoulli.Template:Sfnp In 1755, the summation symbol Σ is attested in Leonhard Euler's Institutiones calculi differentialis.Template:Sfnp<ref>Template:Cite book</ref> Euler uses the symbol in expressions like <math display="inline"> \sum (2wx + w^2) = x^2</math>. The usage of sigma notation was later attested by mathematicians such as Lagrange, who denoted <math display="inline"> \sum </math> and <math display="inline"> \sum ^n </math> in 1772.Template:Sfnp<ref>Template:Cite book</ref> Fourier and C. G. J. Jacobi also denoted the sigma notation in 1829,Template:Sfnp but Fourier included lower and upper bounds as in <math display="inline">\sum_{i=1}^{\infty}e^{-i^2t} \ldots</math>.<ref>Template:Cite book</ref><ref>Template:Cite book</ref> Other than sigma notation, the capital letter S is attested as a summation symbol for series in 1823, which was apparently widespread.Template:Sfnp

Formal definitionEdit

Summation may be defined recursively as follows:

<math>\sum_{i=a}^b g(i)=0</math>, for <math>b<a </math>;

<math>\sum_{i=a}^b g(i)=g(b)+\sum_{i=a}^{b-1} g(i)</math>, for <math>b \geqslant a</math>.

Measure theory notationEdit

In the notation of measure and integration theory, a sum can be expressed as a definite integral,

<math>\sum_{k \mathop =a}^b f(k) = \int_{[a,b]} f\,d\mu</math>

where <math>[a, b]</math> is the subset of the integers from <math>a</math> to <math>b</math>, and where <math>\mu</math> is the counting measure over the integers.

Calculus of finite differencesEdit

Given a function Template:Mvar that is defined over the integers in the interval Template:Math, the following equation holds:

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:

where

is the derivative of Template:Mvar.

An example of application of the above equation is the following:

<math>n^k=\sum_{i=0}^{n-1} \left((i+1)^k-i^k\right).</math>

Using binomial theorem, this may be rewritten as:

<math>n^k=\sum_{i=0}^{n-1} \biggl(\sum_{j=0}^{k-1} \binom{k}{j} i^j\biggr).</math>

The above formula is more commonly used for inverting of the difference operator <math>\Delta</math>, defined by:

<math>\Delta(f)(n)=f(n+1)-f(n), </math>

where Template:Mvar is a function defined on the nonnegative integers. Thus, given such a function Template:Mvar, the problem is to compute the antidifference of Template:Mvar, a function <math>F=\Delta^{-1}f</math> such that <math>\Delta F=f</math>. That is, <math>F(n+1)-F(n)=f(n).</math> This function is defined up to the addition of a constant, and may be chosen as<ref name=CRC>Handbook of Discrete and Combinatorial Mathematics, Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, Template:Isbn.</ref>

There is not always a closed-form expression for such a summation, but Faulhaber's formula provides a closed form in the case where <math>f(n)=n^k</math> and, by linearity, for every polynomial function of Template:Mvar.

Approximation by definite integralsEdit

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

<math>\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,</math>

since the right-hand side is by definition the limit for <math>n\to\infty</math> 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.

IdentitiesEdit

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 identitiesEdit

<math>\sum_{n=s}^t C\cdot f(n) = C\cdot \sum_{n=s}^t f(n) \quad</math> (distributivity)<ref name="vpr">Template:Cite book</ref>

<math>\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</math> (commutativity and associativity)<ref name="vpr"/>

<math>\sum_{n=s}^t f(n) = \sum_{n=s+p}^{t+p} f(n-p)\quad</math> (index shift)

<math>\sum_{n\in B} f(n) = \sum_{m\in A} f(\sigma(m)), \quad</math> for a bijection Template:Mvar from a finite set Template:Mvar onto a set Template:Mvar (index change); this generalizes the preceding formula.

<math>\sum_{n=s}^t f(n) =\sum_{n=s}^j f(n) + \sum_{n=j+1}^t f(n)\quad</math> (splitting a sum, using associativity)

<math>\sum_{n=a}^{b}f(n)=\sum_{n=0}^{b}f(n)-\sum_{n=0}^{a-1}f(n)\quad</math> (a variant of the preceding formula)

<math>\sum_{n=s}^t f(n) = \sum_{n=0}^{t-s} f(t-n)\quad</math> (the sum from the first term up to the last is equal to the sum from the last down to the first)

<math>\sum_{n=0}^t f(n) = \sum_{n=0}^{t} f(t-n)\quad</math> (a particular case of the formula above)

<math>\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</math> (commutativity and associativity, again)

<math>\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</math> (another application of commutativity and associativity)

<math>\sum_{n=2s}^{2t+1} f(n) = \sum_{n=s}^t f(2n) + \sum_{n=s}^t f(2n+1)\quad</math> (splitting a sum into its odd and even parts, for even indexes)

<math>\sum_{n=2s+1}^{2t} f(n) = \sum_{n=s+1}^t f(2n) + \sum_{n=s+1}^t f(2n-1)\quad</math> (splitting a sum into its odd and even parts, for odd indexes)

<math>\biggl(\sum_{i=0}^{n} a_i\biggr) \biggl(\sum_{j=0}^{n} b_j\biggr)=\sum_{i=0}^n \sum_{j=0}^n a_ib_j \quad</math> (distributivity)

<math>\sum_{i=s}^m\sum_{j=t}^n {a_i}{c_j} = \biggl(\sum_{i=s}^m a_i\biggr) \biggl( \sum_{j=t}^n c_j \biggr)\quad</math> (distributivity allows factorization)

<math>\sum_{n=s}^t \log_b f(n) = \log_b \prod_{n=s}^t f(n)\quad</math> (the logarithm of a product is the sum of the logarithms of the factors)

<math>C^{\sum\limits_{n=s}^t f(n) } = \prod_{n=s}^t C^{f(n)}\quad</math> (the exponential of a sum is the product of the exponential of the summands)

<math>\sum^{k}_{m = 0}\sum^{m}_{n = 0}f(m,n)=\sum^{k}_{m = 0}\sum^{k}_{n = m}f(n,m),\quad</math>for any function <math display="inline">f</math> from <math display="inline">\mathbb{Z}\times\mathbb{Z}</math>.

Powers and logarithm of arithmetic progressionsEdit

<math>\sum_{i=1}^n c = nc\quad</math> for every Template:Mvar that does not depend on Template:Mvar

<math>\sum_{i=0}^n i = \sum_{i=1}^n i = \frac{n(n+1)}{2}\qquad</math> (Sum of the simplest arithmetic progression, consisting of the first n natural numbers.)Template:R

<math>\sum_{i=1}^n (2i-1) = n^2\qquad</math> (Sum of first odd natural numbers)

<math>\sum_{i=0}^{n} 2i = n(n+1)\qquad</math> (Sum of first even natural numbers)

<math>\sum_{i=1}^{n} \log i = \log (n!)\qquad</math> (A sum of logarithms is the logarithm of the product)

<math>\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</math> (Sum of the first squares, see square pyramidal number.) Template:R

<math>\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</math> (Nicomachus's theorem) Template:R

More generally, one has Faulhaber's formula for <math>p>1</math>

<math> \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},</math>

where <math>B_k</math> denotes a Bernoulli number, and <math>\binom p k</math> is a binomial coefficient.

Summation index in exponentsEdit

In the following summations, Template:Mvar is assumed to be different from 1.

<math>\sum_{i=0}^{n-1} a^i = \frac{1-a^n}{1-a}</math> (sum of a geometric progression)

<math>\sum_{i=0}^{n-1} \frac{1}{2^i} = 2-\frac{1}{2^{n-1}}</math> (special case for Template:Math)

<math>\sum_{i=0}^{n-1} i a^i =\frac{a-na^n+(n-1)a^{n+1}}{(1-a)^2}</math> (Template:Mvar times the derivative with respect to Template:Mvar of the geometric progression)

<math>\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}</math>

(sum of an arithmetico–geometric sequence)

Binomial coefficients and factorialsEdit

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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 theoremEdit

<math>\sum_{i=0}^n {n \choose i}a^{n-i} b^i=(a + b)^n,</math> the binomial theorem

<math>\sum_{i=0}^n {n \choose i} = 2^n,</math> the special case where Template:Math

<math>\sum_{i=0}^n {n \choose i}p^i (1-p)^{n-i}=1</math>, the special case where Template:Math, which, for <math>0 \le p \le 1,</math> expresses the sum of the binomial distribution

<math>\sum_{i=0}^{n} i{n \choose i} = n(2^{n-1}),</math> the value at Template:Math of the derivative with respect to Template:Mvar of the binomial theorem

<math>\sum_{i=0}^n \frac{n \choose i}{i+1} = \frac{2^{n+1}-1}{n+1},</math> the value at Template:Math of the antiderivative with respect to Template:Mvar of the binomial theorem

Involving permutation numbersEdit

In the following summations, <math>{}_{n}P_{k}</math> is the number of [[k-permutation|Template:Math-permutations of Template:Math]].

<math>\sum_{i=0}^{n} {}_{i}P_{k}{n \choose i} = {}_{n}P_{k}(2^{n-k})</math>

<math>\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}^+</math>, where and <math>\lfloor x\rfloor</math> denotes the floor function.

OthersEdit

<math>\sum_{k=0}^{m} \binom{n+k}{n} = \binom{n+m+1}{n+1}</math>

<math>\sum_{i=k}^{n} {i \choose k} = {n+1 \choose k+1}</math>

<math>\sum_{i=0}^n {m+i-1 \choose i} = {m+n \choose n}</math>

<math>\sum_{i=0}^n {n \choose i}^2 = {2n \choose n}</math>

<math>\sum_{i=0}^n \frac{1}{i!} = \frac{\lfloor n!\; e \rfloor}{n!}</math>

Harmonic numbersEdit

<math>\sum_{i=1}^n \frac{1}{i} = H_n\quad</math> (the Template:Mvarth harmonic number)

<math>\sum_{i=1}^n \frac{1}{i^k} = H^k_n\quad</math> (a generalized harmonic number)

Growth ratesEdit

The following are useful approximations (using theta notation):

<math>\sum_{i=1}^n i^c \in \Theta(n^{c+1})</math> for real c greater than −1

<math>\sum_{i=1}^n \frac{1}{i} \in \Theta(\log_e n)</math> (See Harmonic number)

<math>\sum_{i=1}^n c^i \in \Theta(c^n)</math> for real c greater than 1

<math>\sum_{i=1}^n \log(i)^c \in \Theta(n \cdot \log(n)^{c})</math> for non-negative real c

<math>\sum_{i=1}^n \log(i)^c \cdot i^d \in \Theta(n^{d+1} \cdot \log(n)^{c})</math> for non-negative real c, d

<math>\sum_{i=1}^n \log(i)^c \cdot i^d \cdot b^i \in \Theta (n^d \cdot \log(n)^c \cdot b^n)</math> for non-negative real b > 1, c, d

NotesEdit

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ReferencesEdit

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BibliographyEdit

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External linksEdit

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