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| {{Calculation results}}
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| '''Summation''' is the operation of [[addition|adding]] a [[sequence]] of numbers; the result is their '''sum''' or ''total''. If numbers are added sequentially from left to right, any intermediate result is a [[partial sum]], [[prefix sum]], or [[running total]] of the summation. The numbers to be summed (called '''addends''', or sometimes '''summands''') may be [[integer]]s, [[rational number]]s, [[real number]]s, or [[complex number]]s. Besides numbers, other types of values can be added as well: [[vector space|vectors]], [[matrix (mathematics)|matrices]], [[polynomial]]s and, in general, elements of any [[Abelian group|additive group]] (or even [[monoid]]). For finite sequences of such elements, summation always produces a well-defined sum (possibly by virtue of the convention for [[empty sum]]s).
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| Summation of an infinite sequence of values can often result in a value, in the sense of a [[limit (mathematics)|limit]] (although sometimes the value may be infinite, and often no value results at all.) Such infinite summations are known as [[series (mathematics)|series]]. Another notion involving limits of finite sums is [[integral|integration]]. The term summation has a special meaning related to [[extrapolation]] in the context of [[divergent series]].
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| The summation of the sequence <nowiki>[</nowiki>1, 2, 4, 2<nowiki>]</nowiki> is an [[expression (mathematics)|expression]] whose value is the sum of each of the members of the sequence. In the example, {{nowrap|1 + 2 + 4 + 2}} = 9. Since addition is [[associative]] the value does not depend on how the additions are grouped, for instance {{nowrap|(1 + 2) + (4 + 2)}} and {{nowrap|1 + ((2 + 4) + 2)}} both have the value 9; therefore, parentheses are usually omitted in repeated additions. Addition is also [[commutative]], so [[permutation|permuting]] the terms of a finite sequence does not change its sum (for infinite summations this property may fail; see [[absolute convergence]] for conditions under which it still holds).
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| There is no special notation for the summation of such explicit sequences, as the corresponding repeated addition expression will do. There is only a slight difficulty if the sequence has fewer than two elements: the summation of a sequence of one term involves no plus sign (it is indistinguishable from the term itself) and the summation of the empty sequence cannot even be written down (but one can write its value "0" in its place). If, however, the terms of the sequence are given by a regular pattern, possibly of variable length, then a summation operator may be useful or even essential. For the summation of the sequence of consecutive integers from 1 to 100 one could use an addition expression involving an [[ellipsis]] to indicate the missing terms: {{nowrap|1 + 2 + 3 + 4 + ... + 99 + 100}}. In this case the reader easily guesses the pattern; however, for more complicated patterns, one needs to be precise about the rule used to find successive terms, which can be achieved by using the summation operator "[[sigma|Σ]]". Using this sigma notation the above summation is written as:
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| :<math>\sum_{i \mathop =1}^{100}i.</math>
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| The value of this summation is 5050. It can be found without performing 99 additions, since it can be shown (for instance by [[mathematical induction]]) that | |
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| :<math>\sum_{ i \mathop =1}^ni = \frac{n(n+1)}{2}</math>
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| for all [[natural numbers]] ''n''. More generally, formulae exist for many summations of terms following a regular pattern.
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| The summation of ''all natural numbers'' to ''infinity'' is '''"minus one-twelfth"'''.<ref name="NYT-20140203">{{cite news |last=Overbye |first=Dennis |authorlink=Dennis Overbye |title=In the End, It All Adds Up to –1/12|url=http://www.nytimes.com/2014/02/04/science/in-the-end-it-all-adds-up-to.html |date=February 3, 2014|work=[[New York Times]] |accessdate=February 3, 2014 }}</ref>
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| :<math>\sum_{n=1}^{\infty} n = - \frac {1}{12}</math>
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| The term "[[indefinite sum]]mation" refers to the search for an [[inverse image]] of a given infinite sequence ''s'' of values for the forward [[difference operator]], in other words for a sequence, called antidifference of ''s'', whose [[finite difference]]s are given by ''s''. By contrast, summation as discussed in this article is called "definite summation".
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| When it is necessary to clarify that numbers are added with their signs, the term '''algebraic sum'''<ref>Oxford English Dictionary, 2nd ed. - algebraic (esp. of a sum): taken with consideration of the sign (plus or minus) of each term.</ref> is used. For example, in electric circuit theory [[Kirchhoff's circuit laws]] consider the algebraic sum of currents in a network of conductors meeting at a point, assigning opposite signs to currents flowing in and out of the node.
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| == Notation ==
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| === Capital-sigma notation ===
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| Mathematical notation uses a symbol that compactly represents summation of many similar terms: the ''summation symbol'', '''[[Sigma|∑]]''', an enlarged form of the upright capital Greek letter [[Sigma (letter)|Sigma]]. This is defined as:
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| :<math>\sum_{i \mathop =m}^n a_i = a_m + a_{m+1} + a_{m+2} +\cdots+ a_{n-1} + a_n. </math>
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| Where, ''i'' represents the '''index of summation'''; ''a<sub>i</sub>'' is an indexed variable representing each successive term in the series; ''m'' is the '''lower bound of summation''', and ''n'' is the '''upper bound of summation'''. The ''"i = m"'' under the summation symbol means that the index ''i'' starts out equal to ''m''. The index, ''i'', is incremented by 1 for each successive term, stopping when ''i'' = ''n''.<ref>For a detailed exposition on summation notation, and arithmetic with sums, see {{cite book|authors=Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren|chapter=Chapter 2: Sums|title=Concrete Mathematics: A Foundation for Computer Science (2nd Edition)|publisher= Addison-Wesley Professional|year=1994|isbn=978-0201558029|url=http://www.cse.iitb.ac.in/~vsevani/Concrete%20Mathematics%20-%20R.%20Graham,%20D.%20Knuth,%20O.%20Patashnik.pdf}}</ref>
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| Here is an example showing the summation of exponential terms (all terms to the power of 2):
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| :<math>\sum_{i \mathop =3}^6 i^2 = 3^2+4^2+5^2+6^2 = 86.</math>
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| Informal writing sometimes omits the definition of the index and bounds of summation when these are clear from context, as in:
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| :<math>\sum a_i^2 = \sum_{ i \mathop =1}^n a_i^2.</math>
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| One often sees generalizations of this notation in which an arbitrary logical condition is supplied, and the sum is intended to be taken over all values satisfying the condition. For example:
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| :<math>\sum_{0\le k< 100} f(k)</math>
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| is the sum of ''f''(''k'') over all (integers) ''k'' in the specified range,
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| :<math>\sum_{x \mathop \in S} f(x)</math>
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| is the sum of ''f''(''x'') over all elements ''x'' in the set ''S'', and
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| :<math>\sum_{d|n}\;\mu(d)</math>
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| is the sum of μ(''d'') over all positive integers ''d'' dividing ''n''.<ref>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 (''i'' through ''q'') 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 ''x'' instead of ''k'' in the above formulae involving ''k''. See also [[typographical conventions in mathematical formulae]].</ref> | |
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| There are also ways to generalize the use of many sigma signs. For example,
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| :<math>\sum_{\ell,\ell'}</math>
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| is the same as
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| :<math>\sum_\ell\sum_{\ell'}.</math>
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| A similar notation is applied when it comes to denoting the [[Multiplication#Capital Pi notation|product]] of a sequence, which is similar to its summation, but which uses the multiplication operation instead of addition (and gives 1 for an empty sequence instead of 0). The same basic structure is used, with ∏, an enlarged form of the Greek capital letter [[Pi (letter)|Pi]], replacing the ∑. | |
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| ===Special cases===
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| It is possible to sum fewer than 2 numbers:
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| * If the summation has one summand ''x'', then the evaluated sum is ''x''.
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| * If the summation has no summands, then the evaluated sum is [[0 (number)|zero]], because <!-- huh?, REFERENCE?? --> zero is the [[identity element|identity]] for addition. This is known as the ''[[empty sum]]''.
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| These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case.
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| 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.
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| == Formal definition ==
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| If the [[iterated function]] notation is defined e.g. <math>f^2(x) \equiv f(f(x))</math> and is considered a more primitive notation, then summation can be defined in terms of iterated functions as:
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| :<math>
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| \left\{b+1,\sum_{i \mathop =a}^b g(i)\right\} \equiv \left( \{i,x\} \rightarrow \{ i+1 ,x+g(i) \}\right)^{b-a+1} \{a,0\}</math>
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| Where the curly braces define a 2-[[tuple]] and the right arrow is a function definition taking a 2-tuple to 2-tuple. The function is applied b-a+1 times on the tuple {a,0}.
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| ==Measure theory notation==
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| In the notation of [[measure theory|measure]] and [[integral|integration]] theory, a sum can be expressed as a [[definite integral]],
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| <math>\sum_{k \mathop =a}^b f(k) = \int_{[a,b]} f\,d\mu</math> | |
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| where {{math|[''a'',''b'']}} is the subset of the [[integer]]s from {{math|''a''}} to {{math|''b''}}, and where {{math|''μ''}} is the [[counting measure]].
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| ==Fundamental theorem of discrete calculus==
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| [[Indefinite sum]]s can be used to calculate definite sums with the formula:<ref>"Handbook of discrete and combinatorial mathematics", Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, ISBN 0-8493-0149-1</ref>
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| :<math>\sum_{k=a}^b f(k)=\Delta^{-1}f(b+1)-\Delta^{-1}f(a)</math>
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| ==Approximation by definite integrals==
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| Many such approximations can be obtained by the following connection between sums and [[integral]]s, which holds for any:
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| [[monotonic function|increasing]] function ''f'':
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| :<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>
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| [[monotonic function|decreasing]] function ''f'':
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| :<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>
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| For more general approximations, see the [[Euler–Maclaurin formula]].
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| 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
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| :<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>
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| 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.
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| == Identities ==
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| 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]]
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| === General manipulations ===
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| : <math>\sum_{n=s}^t C\cdot f(n) = C\cdot \sum_{n=s}^t f(n)</math>, where ''C'' is a constant
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| : <math>\sum_{n=s}^t f(n) + \sum_{n=s}^{t} g(n) = \sum_{n=s}^t \left[f(n) + g(n)\right]</math>
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| : <math>\sum_{n=s}^t f(n) - \sum_{n=s}^{t} g(n) = \sum_{n=s}^t \left[f(n) - g(n)\right]</math> | |
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| : <math>\sum_{n=s}^t f(n) = \sum_{n=s+p}^{t+p} f(n-p)</math>
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| : <math>\sum_{n=s}^j f(n) + \sum_{n=j+1}^t f(n) = \sum_{n=s}^t f(n)</math>
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| : <math>\sum_{n\in A} f(n) = \sum_{n\in \sigma(A)} f(n)</math>, for a finite set '''A''' (Where σ(A) is a [[permutation]] of A).
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| : <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}</math>
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| : <math>\sum_{n=0}^t f(2n) + \sum_{n=0}^t f(2n+1) = \sum_{n=0}^{2t+1} f(n)</math>
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| : <math>\sum_{n=0}^t \sum_{i=0}^{z-1} f(z\cdot n+i) = \sum_{n=0}^{z\cdot t+z-1} f(n)</math>
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| : <math>\sum_{n=s}^t \ln f(n) = \ln \prod_{n=s}^t f!(n)</math>
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| : <math>c^{\left[\sum_{n=s}^t f(n) \right]} = \prod_{n=s}^t c^{f(n)}</math>
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| === Some summations of polynomial expressions ===
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| : <math>\sum_{i=m}^n 1 = n+1-m</math>
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| : <math>\sum_{i=1}^n \frac{1}{i} = H_n</math> (See [[Harmonic number]])
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| : <math>\sum_{i=1}^n \frac{1}{i^k} = H^k_n</math> (See [[Harmonic number#Generalized harmonic numbers|Generalized harmonic number]])
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| : <math>\sum_{i=m}^n i = \frac{n(n+1)}{2} - \frac{m(m-1)}{2} = \frac{(n+1-m)(n+m)}{2}</math> (see [[arithmetic series]])
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| : <math>\sum_{i=0}^n i = \sum_{i=1}^n i = \frac{n(n+1)}{2}</math> (Special case of the arithmetic series)
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| : <math>\sum_{i=0}^n i^2 = \frac{n(n+1)(2n+1)}{6} = \frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6}</math>
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| : <math>\sum_{i=0}^n i^3 = \left(\frac{n(n+1)}{2}\right)^2 = \frac{n^4}{4} + \frac{n^3}{2} + \frac{n^2}{4} = \left[\sum_{i=1}^n i\right]^2</math>
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| : <math>\sum_{i=0}^n i^4 = \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30} = \frac{n^5}{5} + \frac{n^4}{2} + \frac{n^3}{3} - \frac{n}{30}</math>
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| : <math>\sum_{i=0}^n i^p = \frac{(n+1)^{p+1}}{p+1} + \sum_{k=1}^p\frac{B_k}{p-k+1}{p\choose k}(n+1)^{p-k+1}</math> where <math>B_k</math> denotes a [[Bernoulli number]] (see [[Faulhaber's formula]])
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| <br />
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| The following formulae are manipulations of <math>\sum_{i=0}^n i^3 = \left(\sum_{i=0}^n i\right)^2</math> generalized to begin a series at any natural number value (i.e., <math>m \in \mathbb{N}</math> ):
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| : <math>\left(\sum_{i=m}^n i\right)^2 = \sum_{i=m}^n ( i^3 - im(m-1) )</math>
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| : <math>\sum_{i=m}^n i^3 = \left(\sum_{i=m}^n i\right)^2 + m(m-1)\sum_{i=m}^n i</math>
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| === Some summations involving exponential terms ===
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| In the summations below ''a'' is a constant not equal to 1
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| : <math>\sum_{i=m}^{n-1} a^i = \frac{a^m-a^n}{1-a}</math> ({{nowrap|''m'' < ''n''}}; see [[geometric series]])
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| : <math>\sum_{i=0}^{n-1} a^i = \frac{1-a^n}{1-a}</math> (geometric series starting at <math>a_1=1</math>)
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| : <math>\sum_{i=0}^{n-1} i a^i = \frac{a-na^n+(n-1)a^{n+1}}{(1-a)^2}</math>
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| : <math>\sum_{i=0}^{n-1} i 2^i = 2+(n-2)2^{n}</math> (special case when ''a'' = 2)
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| : <math>\sum_{i=0}^{n-1} \frac{i}{2^i} = 2-\frac{n+1}{2^{n-1}}</math> (special case when ''a'' = 1/2)
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| === Some summations involving binomial coefficients and factorials ===
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| There exist enormously 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.
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| : <math>\sum_{i=0}^n {n \choose i} = 2^n</math>
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| : <math>\sum_{i=1}^{n} i{n \choose i} = n2^{n-1}</math>
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| : <math>\sum_{i=0}^{n} i!\cdot{n \choose i} = \sum_{i=0}^{n} {}_{n}P_{i} = \lfloor n!\cdot e \rfloor</math>
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| : <math>\sum_{i=0}^{n-1} {i \choose k} = {n \choose k+1}</math>
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| : <math>\sum_{i=0}^n {n \choose i}a^{(n-i)} b^i=(a + b)^n</math>, the [[binomial theorem]]
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| : <math>\sum_{i=0}^n i\cdot i! = (n+1)! - 1</math>
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| : <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>
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| : <math>\sum_{i=0}^n {m+i-1 \choose i} = {m+n \choose n}</math>
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| ==Growth rates==
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| The following are useful [[approximation]]s (using [[big O notation|theta notation]]):
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| : <math>\sum_{i=1}^n i^c \in \Theta(n^{c+1})</math> for real ''c'' greater than −1
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| : <math>\sum_{i=1}^n \frac{1}{i} \in \Theta(\log n)</math> (See [[Harmonic number]])
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| : <math>\sum_{i=1}^n c^i \in \Theta(c^n)</math> for real ''c'' greater than 1
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| : <math>\sum_{i=1}^n \log(i)^c \in \Theta(n \cdot \log(n)^{c})</math> for [[non-negative]] real ''c''
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| : <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''
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| : <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''
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| ==See also==
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| * [[Checksum]]
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| * [[Einstein notation]]
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| * [[Iterated binary operation]]
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| * [[Kahan summation algorithm]]
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| * [[Product (mathematics)]]
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| ==Notes==
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| {{reflist}}
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| ==Further reading==
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| * [[Nicholas J. Higham]], "[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.3535 The accuracy of floating point summation]", ''SIAM J. Scientific Computing'' '''14''' (4), 783–799 (1993).
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| ==External links==
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| * {{commonscat-inline}}
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| * {{planetmath reference|id=6361|title=Summation}}
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| * [http://upload.wikimedia.org/wikipedia/commons/6/62/Sum_of_i.pdf Derivation of Polynomials to Express the Sum of Natural Numbers with Exponents]
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| * {{cite web|last=Moriarty|first=Philip|title=∑ – Summation (and Fourier Analysis)|url=http://www.sixtysymbols.com/videos/summation.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|coauthors=Bowley, Roger|year=2009}}
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| [[Category:Arithmetic]]
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| [[Category:Mathematical notation]]
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