Summation

This article is about sums of several elements. For more elementary aspects, seeAddition.For infinite sums, seeSeries (mathematics).For other uses, seeSummation (disambiguation).

Inmathematics,summationis theadditionof asequenceofnumbers,calledaddendsorsummands;the result is theirsumortotal.Beside numbers, other types of values can be summed as well:functions,vectors,matrices,polynomialsand, in general, elements of any type ofmathematical objectson which anoperationdenoted "+" is defined.

Summations ofinfinite sequencesare calledseries.They involve the concept oflimit,and are not considered in this article.

The summation of an explicit sequence is denoted as a succession of additions. For example, summation of[1, 2, 4, 2]is denoted1 + 2 + 4 + 2,and results in 9, that is,1 + 2 + 4 + 2 = 9.Because addition isassociativeandcommutative,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 element results in this element 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 afunctionof 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 100natural numbersmay be written as1 + 2 + 3 + 4 + ⋯ + 99 + 100.Otherwise, summation is denoted by usingΣ notation,where${\textstyle \sum }$is an enlarged capitalGreek lettersigma.For example, the sum of the firstnnatural numbers can be denoted as${\textstyle \sum _{i=1}^{n}i.}$

For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to findclosed-form expressionsfor the result. For example,

$\sum _{i=1}^{n}i={\frac {n(n+1)}{2}}.$

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.

Notation

Further information:Iterated binary operation § Notation

Capital-sigma notation

The summation symbol

Mathematical notation uses a symbol that compactly represents summation of many similar terms: thesummation symbol,${\textstyle \sum }$,an enlarged form of the upright capital Greek lettersigma.This is defined as

$\sum _{i\mathop {=} m}^{n}a_{i}=a_{m}+a_{m+1}+a_{m+2}+\cdots +a_{n-1}+a_{n}$

whereiis theindex of summation;a_iis an indexed variable representing each term of the sum;mis thelower bound of summation,andnis theupper bound of summation.The "i=m"under the summation symbol means that the indexistarts out equal tom.The index,i,is incremented by one for each successive term, stopping wheni=n.

This is read as "sum ofa_i,fromi=mton".

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.

Alternatively, 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 ton.For example, one might write that:

$\sum a_{i}^{2}=\sum _{i=1}^{n}a_{i}^{2}.$

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:

$\sum _{0\leq k<100}f(k)$

is an alternative notation for${\textstyle \sum _{k=0}^{99}f(k),}$the sum of$f(k)$over all (integers)$k$in the specified range. Similarly,

$\sum _{x\mathop {\in } S}f(x)$

is the sum of$f(x)$over all elements$x$in the set$S$,and

$\sum _{d\,|\,n}\;\mu (d)$

is the sum of$\mu (d)$over all positive integers$d$dividing$n$.

There are also ways to generalize the use of many sigma signs. For example,

$\sum _{i,j}$

is the same as

$\sum _{i}\sum _{j}.$

A similar notation is used for theproduct of a sequence,where${\textstyle \prod }$,an enlarged form of the Greek capital letterpi,is used instead of${\textstyle \sum.}$

Special cases

It is possible to sum fewer than 2 numbers:

• If the summation has one summand$x$,then the evaluated sum is$x$.
• If the summation has no summands, then the evaluated sum iszero,because zero is theidentityfor addition. This is known as theempty sum.

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

Algebraic sum

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.

Formal definition

Summation may be defined recursively as follows:

$\sum _{i=a}^{b}g(i)=0$,for$b<a$;

$\sum _{i=a}^{b}g(i)=g(b)+\sum _{i=a}^{b-1}g(i)$,for$b\geqslant a$.

Measure theory notation

In the notation ofmeasureandintegrationtheory, a sum can be expressed as adefinite integral,

$\sum _{k\mathop {=} a}^{b}f(k)=\int _{[a,b]}f\,d\mu$

where$[a,b]$is thesubsetof the integers from$a$to$b$,and where$\mu$is thecounting measureover the integers.

Calculus of finite differences

Given a functionfthat is defined over the integers in theinterval[m,n],the following equation holds:

$f(n)-f(m)=\sum _{i=m}^{n-1}(f(i+1)-f(i)).$

This is known as atelescoping seriesand is the analogue of thefundamental theorem of calculusincalculus 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 thederivativeoff.

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

$n^{k}=\sum _{i=0}^{n-1}\left((i+1)^{k}-i^{k}\right).$

Usingbinomial 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 thedifference operator$\Delta$,defined by:

$\Delta (f)(n)=f(n+1)-f(n),$

wherefis a function defined on the nonnegative integers. Thus, given such a functionf,the problem is to compute theantidifferenceoff,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 aclosed-form expressionfor such a summation, butFaulhaber's formulaprovides a closed form in the case where$f(n)=n^{k}$and, bylinearity,for everypolynomial functionofn.

Approximation by definite integrals

Many such approximations can be obtained by the following connection between sums andintegrals,which holds for anyincreasingfunctionf:

$\int _{s=a-1}^{b}f(s)\ ds\leq \sum _{i=a}^{b}f(i)\leq \int _{s=a}^{b+1}f(s)\ ds.$

and for anydecreasingfunctionf:

$\int _{s=a}^{b+1}f(s)\ ds\leq \sum _{i=a}^{b}f(i)\leq \int _{s=a-1}^{b}f(s)\ ds.$

For more general approximations, see theEuler–Maclaurin formula.

For summations in which the summand is given (or can be interpolated) by anintegrablefunction of the index, the summation can be interpreted as aRiemann sumoccurring 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 summationnis fixed, and little can be said about the error in the above approximation without additional assumptions aboutf: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 involvingtrigonometric functionsor othertranscendental functions,seelist of mathematical series.

General identities

$\sum _{n=s}^{t}C\cdot f(n)=C\cdot \sum _{n=s}^{t}f(n)\quad$(distributivity)

$\sum _{n=s}^{t}f(n)\pm \sum _{n=s}^{t}g(n)=\sum _{n=s}^{t}\left(f(n)\pm g(n)\right)\quad$(commutativityandassociativity)

$\sum _{n=s}^{t}f(n)=\sum _{n=s+p}^{t+p}f(n-p)\quad$(index shift)

$\sum _{n\in B}f(n)=\sum _{m\in A}f(\sigma (m)),\quad$for abijectionσfrom a finite setAonto a setB(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, usingassociativity)

$\sum _{n=a}^{b}f(n)=\sum _{n=0}^{b}f(n)-\sum _{n=0}^{a-1}f(n)\quad$(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)

$\sum _{k\leq j\leq i\leq n}a_{i,j}=\sum _{i=k}^{n}\sum _{j=k}^{i}a_{i,j}=\sum _{j=k}^{n}\sum _{i=j}^{n}a_{i,j}=\sum _{j=0}^{n-k}\sum _{i=k}^{n-j}a_{i+j,i}\quad$(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 itsoddandevenparts, 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)

${\biggl (}\sum _{i=0}^{n}a_{i}{\biggr )}{\biggl (}\sum _{j=0}^{n}b_{j}{\biggr )}=\sum _{i=0}^{n}\sum _{j=0}^{n}a_{i}b_{j}\quad$(distributivity)

$\sum _{i=s}^{m}\sum _{j=t}^{n}{a_{i}}{c_{j}}={\biggl (}\sum _{i=s}^{m}a_{i}{\biggr )}{\biggl (}\sum _{j=t}^{n}c_{j}{\biggr )}\quad$(distributivity allows factorization)

$\sum _{n=s}^{t}\log _{b}f(n)=\log _{b}\prod _{n=s}^{t}f(n)\quad$(thelogarithmof a product is the sum of the logarithms of the factors)

$C^{\sum \limits _{n=s}^{t}f(n)}=\prod _{n=s}^{t}C^{f(n)}\quad$(theexponentialof a sum is the product of the exponential of the summands)

$\sum _{m=0}^{k}\sum _{n=0}^{m}f(m,n)=\sum _{m=0}^{k}\sum _{n=m}^{k}f(n,m),\quad$for any function${\textstyle f}$from${\textstyle \mathbb {Z} \times \mathbb {Z} }$.

Powers and logarithm of arithmetic progressions

$\sum _{i=1}^{n}c=nc\quad$for everycthat does not depend oni

$\sum _{i=0}^{n}i=\sum _{i=1}^{n}i={\frac {n(n+1)}{2}}\qquad$(Sum of the simplestarithmetic progression,consisting of the firstnnatural 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 oflogarithmsis 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 firstsquares,seesquare 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 hasFaulhaber's formulafor$p>1$

$\sum _{k=1}^{n}k^{p}={\frac {n^{p+1}}{p+1}}+{\frac {1}{2}}n^{p}+\sum _{k=2}^{p}{\binom {p}{k}}{\frac {B_{k}}{p-k+1}}\,n^{p-k+1},$

where$B_{k}$denotes aBernoulli number,and${\binom {p}{k}}$is abinomial coefficient.

Summation index in exponents

In the following summations,ais assumed to be different from 1.

$\sum _{i=0}^{n-1}a^{i}={\frac {1-a^{n}}{1-a}}$(sum of ageometric progression)

$\sum _{i=0}^{n-1}{\frac {1}{2^{i}}}=2-{\frac {1}{2^{n-1}}}$(special case fora= 1/2)

$\sum _{i=0}^{n-1}ia^{i}={\frac {a-na^{n}+(n-1)a^{n+1}}{(1-a)^{2}}}$(atimes the derivative with respect toaof the geometric progression)

${\begin{aligned}\sum _{i=0}^{n-1}\left(b+id\right)a^{i}&=b\sum _{i=0}^{n-1}a^{i}+d\sum _{i=0}^{n-1}ia^{i}\\&=b\left({\frac {1-a^{n}}{1-a}}\right)+d\left({\frac {a-na^{n}+(n-1)a^{n+1}}{(1-a)^{2}}}\right)\\&={\frac {b(1-a^{n})-(n-1)da^{n}}{1-a}}+{\frac {da(1-a^{n-1})}{(1-a)^{2}}}\end{aligned}}$

(sum of anarithmetico–geometric sequence)

Binomial coefficients and factorials

Main article:Binomial coefficient § Sums of the binomial coefficients

There exist very many summation identities involving binomial coefficients (a whole chapter ofConcrete Mathematicsis devoted to just the basic techniques). Some of the most basic ones are the following.

Involving the binomial theorem

$\sum _{i=0}^{n}{n \choose i}a^{n-i}b^{i}=(a+b)^{n},$thebinomial theorem

$\sum _{i=0}^{n}{n \choose i}=2^{n},$the special case wherea=b= 1

$\sum _{i=0}^{n}{n \choose i}p^{i}(1-p)^{n-i}=1$,the special case wherep=a= 1 −b,which, for$0\leq p\leq 1,$expresses the sum of thebinomial distribution

$\sum _{i=0}^{n}i{n \choose i}=n(2^{n-1}),$the value ata=b= 1of thederivativewith respect toaof the binomial theorem

$\sum _{i=0}^{n}{\frac {n \choose i}{i+1}}={\frac {2^{n+1}-1}{n+1}},$the value ata=b= 1of theantiderivativewith respect toaof the binomial theorem

Involving permutation numbers

In the following summations,${}_{n}P_{k}$is the number ofk-permutations ofn.

$\sum _{i=0}^{n}{}_{i}P_{k}{n \choose i}={}_{n}P_{k}(2^{n-k})$

$\sum _{i=1}^{n}{}_{i+k}P_{k+1}=\sum _{i=1}^{n}\prod _{j=0}^{k}(i+j)={\frac {(n+k+1)!}{(n-1)!(k+2)}}$

$\sum _{i=0}^{n}i!\cdot {n \choose i}=\sum _{i=0}^{n}{}_{n}P_{i}=\lfloor n!\cdot e\rfloor,\quad n\in \mathbb {Z} ^{+}$,where and$\lfloor x\rfloor$denotes thefloor 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

$\sum _{i=1}^{n}{\frac {1}{i}}=H_{n}\quad$(thenthharmonic number)

$\sum _{i=1}^{n}{\frac {1}{i^{k}}}=H_{n}^{k}\quad$(ageneralized harmonic number)

Growth rates

The following are usefulapproximations(usingtheta notation):

$\sum _{i=1}^{n}i^{c}\in \Theta (n^{c+1})$for realcgreater than −1

$\sum _{i=1}^{n}{\frac {1}{i}}\in \Theta (\log _{e}n)$(SeeHarmonic number)

$\sum _{i=1}^{n}c^{i}\in \Theta (c^{n})$for realcgreater than 1

$\sum _{i=1}^{n}\log(i)^{c}\in \Theta (n\cdot \log(n)^{c})$fornon-negativerealc

$\sum _{i=1}^{n}\log(i)^{c}\cdot i^{d}\in \Theta (n^{d+1}\cdot \log(n)^{c})$for non-negative realc,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 realb> 1,c,d

History

• In 1675,Gottfried Wilhelm Leibniz,in a letter toHenry Oldenburg,suggests the symbol ∫ to mark the sum of differentials (Latin:calculus summatorius), hence the S-shape.The renaming of this symbol tointegralarose later in exchanges withJohann Bernoulli.
• In 1755, the summation symbol Σ is attested inLeonhard Euler'sInstitutiones calculi differentialis.Euler uses the symbol in expressions like:

$\Sigma \ (2wx+w^{2})=x^{2}$

• In 1772, usage of Σ and Σⁿis attested byLagrange.
• In 1823, the capital letterSis attested as a summation symbol for series. This usage was apparently widespread.
• In 1829, the summation symbol Σ is attested byFourierandC. G. J. Jacobi.Fourier's use includes lower and upper bounds, for example:

$\sum _{i=1}^{\infty }e^{-i^{2}t}\ldots$

See also

• Capital-pi notation
• Einstein notation
• Iverson bracket
• Iterated binary operation
• Kahan summation algorithm
• Product (mathematics)
• Summation by parts
• ∑the summation single glyph (U+2211N-ARY SUMMATION)
• ⎲the paired glyph's beginning (U+23B2SUMMATION TOP)
• ⎳the paired glyph's end (U+23B3SUMMATION BOTTOM)

Bibliography

• Cajori, Florian(1929).A History Of Mathematical Notations Volume II.Open Court Publishing.ISBN978-0-486-67766-8.

External links

• Media related toSummationat Wikimedia Commons