Editing Summation

{{Short description|Addition of several numbers or other values}}
{{About|sums of several elements|more elementary aspects|Addition|infinite sums|Series (mathematics)|other uses}}

In [[mathematics]], '''summation''' is the [[addition]] of a [[sequence]] of [[number]]s, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: [[function (mathematics)|functions]], [[vector space|vectors]], [[matrix (mathematics)|matrices]], [[polynomial]]s and, in general, elements of any type of [[mathematical object]]s on which an [[operation (mathematics)|operation]] denoted "+" is defined.

Summations of [[infinite sequence]]s are called [[series (mathematics)|series]]. They involve the concept of [[limit (mathematics)|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 {{math|[1, 2, 4, 2]}} is denoted {{math|1 + 2 + 4 + 2}}, and results in 9, that is, {{math|1=1 + 2 + 4 + 2 = 9}}. 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 (mathematics)|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 number]]s may be written as {{math|1 + 2 + 3 + 4 + ⋯ + 99 + 100}}. Otherwise, summation is denoted by using [[#Capital-sigma notation|Σ notation]], where <math display="inline">\sum</math> is an enlarged capital [[Greek letter]] [[sigma]]. For example, the sum of the first {{mvar|n}} natural numbers can be denoted as

:<math>\sum_{i=1}^n i</math>

For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find [[closed-form expression]]s for the result. For example,{{efn|For details, see [[Triangular number]].}}

:<math>\sum_{i=1}^n i = \frac{n(n+1)}{2}.</math>

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 ===
[[File:Sigma summation notation.svg|thumb|An explanation of the sigma (Σ) summation notation|class=skin-invert-image]]
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>{{cite book
 | last = Apostol | first = Tom M.
 | title = Calculus
 | year = 1967
 | publisher = [[John Wiley & Sons]]
 | isbn = 0-471-00005-1
 | edition = 2nd
 | volume = 1
 | location = USA
 | pages = 37
}}</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 {{math|''i''}} is the "index of summation" or "dummy variable"{{sfnp|Koshy|2002|p=[https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA10 10]}}, {{math|''a<sub>i</sub>''}} is an indexed variable representing each term of the sum; {{math|''m''}} is the "lower bound of summation", and {{math|''n''}} is the "upper bound of summation". The "{{math|1=''i'' = ''m''}}" under the summation symbol means that the index {{math|''i''}} starts out equal to {{math|''m''}}. The index, {{math|''i''}}, is incremented by one for each successive term, stopping when {{math|1=''i'' = ''n''}}.{{efn|For a detailed exposition on summation notation, and arithmetic with sums, see {{cite book
 | last1 = Graham | first1 = Ronald L.
 | last2 = Knuth | first2 = Donald E.
 | last3 = Patashnik | first3 = Oren
 | year = 1994
 | title = Concrete Mathematics: A Foundation for Computer Science
 | edition = 2nd
 | chapter = Chapter 2: Sums
 | publisher = Addison-Wesley Professional
 | isbn = 978-0201558029
}}}} This is read as "sum of {{math|''a<sub>i</sub>''}}, from {{math|1=''i'' = ''m''}} to {{math|''n''}}". 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.{{sfnp|Koshy|2002|p=[https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA9 9]}} 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>.{{sfnp|Vivaldi|2014|p=[https://books.google.com/books?id=wpQvBQAAQBAJ&pg=PA34 34]}} 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>,{{efn|In contexts where there is no possibility of confusion with the [[imaginary unit]] <math>i</math>}} <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>{{Cite web|title=Summation Notation|url=http://www.columbia.edu/itc/sipa/math/summation.html|access-date=2020-08-16|website=www.columbia.edu}}</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 ([[integer]]s) <math>k</math> in the specified range.<ref name="franco">{{cite book
 | title = Mathematical Writing
 | first = Franco | last = Vivaldi
 | url = https://books.google.com/books?id=wpQvBQAAQBAJ&pg=PA35
 | page = 35
 | publisher = Springer
 | doi = 10.1007/978-1-4471-6527-9
 | year = 2014
}}</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>{{cite book
 | last = Miller | first = Victor S.
 | title = Handbook of Discrete and Combinatorial Mathematics
 | chapter = Finite Sums and Summation
 | editor-first = Kenneth H. | editor-last = Rosen
 | url = http://books.google.com/books?id=Xj4PEAAAQBAJ&pg=PA196
 | page = 196
}}</ref><ref>{{cite book
 | title = Elementary Number Theory with Applications
 | first = Thomas | last = Koshy
 | year = 2002
 | url = https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA12
 | page = 12
 | publisher = [[Harcourt (publisher)|Harcourt]]
}}</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> [[divisor|dividing]] <math>n</math>.{{efn|Although the name of the [[Free variables and bound variables|dummy variable]] does not matter (by definition), one usually uses letters from the middle of the alphabet (<math>i</math> through <math>q</math>) to denote integers, if there is a risk of confusion. For example, even if there should be no doubt about the interpretation, it could look slightly confusing to many mathematicians to see <math>x</math> instead of <math>k</math> in the above formulae involving <math>k</math>.}}

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>.{{sfnp|Vivaldi|2014|p=[https://books.google.com/books?id=wpQvBQAAQBAJ&pg=PA36 36]}}

The term '''{{vanchor|finite series}}''' is sometimes used when discussing the summation presented above. Contrast to the [[Series (mathematics)|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>.{{sfnp|Vivaldi|2014|p=[https://books.google.com/books?id=wpQvBQAAQBAJ&pg=PA36 36]}}

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 (letter)|pi]], is used instead of <math display="inline">\sum</math>.{{sfnp|Koshy|2002|p=[https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA13 13]}}

===Special cases===
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 [[0 (number)|zero]], because zero is the [[identity element|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 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. e.g.
+1 −1

=== History ===
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>{{cite book
 |first=David M. |last=Burton
 |title=The History of Mathematics: An Introduction
 |year=2011
 |edition=7th
 |publisher=McGraw-Hill
 |isbn=978-0-07-338315-6
 |page=414
}}</ref><ref>{{cite book
 |first=Gottfried Wilhelm |last=Leibniz |author-link=Gottfried Wilhelm Leibniz
 |title=Der Briefwechsel von Gottfried Wilhelm Leibniz mit Mathematikern. Erster Band
 |url=http://name.umdl.umich.edu/AAX2762.0001.001
 |year=1899
 |editor-first=Karl Immanuel
 |editor-last=Gerhardt
 |place=Berlin
 |publisher=Mayer & Müller
 |page=[https://quod.lib.umich.edu/u/umhistmath/aax2762.0001.001/185?page=root;size=100;view=image 154]
}}</ref>{{sfnp|Cajori|1929|pages=[https://archive.org/details/in.ernet.dli.2015.88254/page/n203 181-182]}} The renaming of this symbol to ''[[integral]]'' arose later in exchanges with [[Johann Bernoulli]].{{sfnp|Cajori|1929|pages=[https://archive.org/details/in.ernet.dli.2015.88254/page/n203 181-182]}} In 1755, the summation symbol Σ is attested in [[Leonhard Euler]]'s ''[[Institutiones calculi differentialis]]''.{{sfnp|Cajori|1929|p=[https://archive.org/details/in.ernet.dli.2015.88254/page/n83 61]}}<ref>{{cite book
 |last1=Euler |first1=Leonhard |author-link=Leonhard Euler
 |title=Institutiones Calculi differentialis
 |date=1755
 |location=Petropolis
 |page=[https://www.digitale-sammlungen.de/en/view/bsb10053431?page=54,55 27]
 |url=https://www.digitale-sammlungen.de/en/view/bsb10053431?page=54,55
 |language=Latin
}}</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.{{sfnp|Cajori|1929|p=[https://archive.org/details/in.ernet.dli.2015.88254/page/n83 61]}}<ref>{{cite book
 |last1=Lagrange
 |first1=Joseph-Louis
 |author-link=Joseph-Louis Lagrange
 |title=Oeuvres de Lagrange. Tome 3
 |date=1867–1892
 |location=Paris
 |page=[https://gallica.bnf.fr/ark:/12148/bpt6k229222d/f452.item 451]
 |url=https://gallica.bnf.fr/ark:/12148/bpt6k229222d/f452.item
 |language=French
}}</ref> [[Joseph Fourier|Fourier]] and [[Carl Gustav Jacob Jacobi|C. G. J. Jacobi]] also denoted the sigma notation in 1829,{{sfnp|Cajori|1929|p=[https://archive.org/details/in.ernet.dli.2015.88254/page/n83 61]}} but Fourier included lower and upper bounds as in <math display="inline">\sum_{i=1}^{\infty}e^{-i^2t} \ldots</math>.<ref>{{cite book
 |title=Mémoires de l'Académie royale des sciences de l'Institut de France pour l'année 1825, tome VIII
 |date=1829
 |publisher=Didot
 |location=Paris
 |pages=[https://books.google.com/books?id=Mpu9XDBOmagC&pg=583 581-622]
 |url=https://books.google.com/books?id=Mpu9XDBOmagC&pg=583
 |language=French
}}</ref><ref>{{cite book
 |last1=Fourier
 |first1=Jean-Baptiste Joseph
 |author-link=Joseph Fourier
 |title=Oeuvres de Fourier. Tome 2
 |date=1888–1890
 |publisher=Gauthier-Villars
 |location=Paris
 |page=[https://gallica.bnf.fr/ark:/12148/bpt6k33707/f154.item 149]
 |url=https://gallica.bnf.fr/ark:/12148/bpt6k33707/f154.item
 |language=French
}}</ref> Other than sigma notation, the capital letter ''S'' is attested as a summation symbol for series in 1823, which was apparently widespread.{{sfnp|Cajori|1929|p=[https://archive.org/details/in.ernet.dli.2015.88254/page/n83 61]}}

== Formal definition ==
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 notation==
In the notation of [[measure theory|measure]] and [[integral|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 differences==

Given a function {{mvar|f}} that is defined over the integers in the [[interval (mathematics)|interval]] {{math|[''m'', ''n'']}}, the following equation holds:
:<math>f(n)-f(m)= \sum_{i=m}^{n-1} (f(i+1)-f(i)).</math>

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:
:<math>f(n)-f(m)=\int_m^n f'(x)\,dx,</math>
where
:<math>f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}</math>
is the [[derivative]] of {{mvar|f}}.

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 {{mvar|f}} is a function defined on the nonnegative integers.
Thus, given such a function {{mvar|f}}, the problem is to compute the [[antidifference]] of {{mvar|f}}, 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, {{isbn|0-8493-0149-1}}.</ref>
:<math>F(n)=\sum_{i=0}^{n-1} f(i).</math>

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 {{mvar|n}}.

==Approximation by definite integrals==
Many such approximations can be obtained by the following connection between sums and [[integral]]s, which holds for any [[monotonic function|increasing]] function ''f'':

:<math>\int_{s=a-1}^{b} f(s)\ ds \le \sum_{i=a}^{b} f(i) \le \int_{s=a}^{b+1} f(s)\ ds.</math>

and for any [[monotonic function|decreasing]] function ''f'':

:<math>\int_{s=a}^{b+1} f(s)\ ds \le \sum_{i=a}^{b} f(i) \le \int_{s=a-1}^{b} f(s)\ ds.</math>

For more general approximations, see the [[Euler–Maclaurin formula]].

For summations in which the summand is given (or can be interpolated) by an [[Riemann integral|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.

== Identities ==
The formulae below involve finite sums; for infinite summations or finite summations of expressions involving [[trigonometric function]]s or other [[transcendental function]]s, see [[list of mathematical series]].

=== General identities ===
: <math>\sum_{n=s}^t C\cdot f(n) = C\cdot \sum_{n=s}^t f(n) \quad</math> ([[distributivity]])<ref name="vpr">{{cite book
 | last1 = Varberg | first1 = Dale E.
 | last2 = Purcell | first2 = Edwin J.
 | last3 = Rigdon | first3 = Steven E.
 | title = Calculus
 | year = 2007
 | publisher = [[Pearson Prentice Hall]]
 | edition = 9th
 | isbn = 978-0131469686
 | page = 217
}}</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]] {{mvar|σ}} from a finite set {{mvar|A}} onto a set {{mvar|B}} (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 [[parity (mathematics)|odd]] and [[parity (mathematics)|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 [[exponentiation|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 progressions ===
: <math>\sum_{i=1}^n c = nc\quad</math> for every {{mvar|c}} that does not depend on {{mvar|i}}
: <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.){{r|CRC|p=52}}
: <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 [[logarithm]]s 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 [[square number|squares]], see [[square pyramidal number]].) {{r|CRC|p=52}}
: <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]]) {{r|CRC|p=52}}

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 exponents ===
In the following summations, {{mvar|a}} 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 {{math|1=''a'' = 1/2}})
: <math>\sum_{i=0}^{n-1} i a^i =\frac{a-na^n+(n-1)a^{n+1}}{(1-a)^2}</math> ({{mvar|a}} times the derivative with respect to {{mvar|a}} 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 factorials ===
{{Main|Binomial coefficient#Sums of the binomial coefficients}}

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====
: <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 {{math|1=''a'' = ''b'' = 1}}
: <math>\sum_{i=0}^n {n \choose i}p^i (1-p)^{n-i}=1</math>, the special case where {{math|1=''p'' = ''a'' = 1 − ''b''}}, 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 {{math|1=''a'' = ''b'' = 1}} of the [[derivative]] with respect to {{mvar|a}} 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 {{math|1=''a'' = ''b'' = 1}} of the [[antiderivative]] with respect to {{mvar|a}} of the binomial theorem

==== Involving permutation numbers====
In the following summations, <math>{}_{n}P_{k}</math> is the number of [[k-permutation|{{math|''k''}}-permutations of {{math|''n''}}]].
: <math>\sum_{i=0}^{n} {}_{i}P_{k}{n \choose i} = {}_{n}P_{k}(2^{n-k})</math>
: <math>\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)}</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]].

====Others====
: <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 i\cdot i! = (n+1)! - 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 numbers===

: <math>\sum_{i=1}^n \frac{1}{i} = H_n\quad</math> (the {{mvar|n}}th [[harmonic number]])
: <math>\sum_{i=1}^n \frac{1}{i^k} = H^k_n\quad</math> (a [[generalized harmonic number]])

==Growth rates==
The following are useful [[approximation]]s (using [[big O notation|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''

==See also==
* [[Capital-pi notation]]
* [[Einstein notation]]
* [[Iverson bracket]]
* [[Iterated binary operation]]
* [[Kahan summation algorithm]]
* [[Product (mathematics)]]
* [[Summation by parts]]
* {{section link|Sigma|Unicode}}

==Notes==
{{notelist}}

==References==
{{Reflist}}

== Bibliography ==
* {{cite book
 |first=Florian
 |last=Cajori
 |author-link=Florian Cajori
 |title=A History Of Mathematical Notations Volume II
 |url=https://archive.org/details/in.ernet.dli.2015.88254
 |year=1929
 |publisher=Open Court Publishing
 |isbn=978-0-486-67766-8
}}

==External links==
* {{commons category-inline}}

{{Authority control}}

[[Category:Mathematical notation]]
[[Category:Addition]]