summation

American Heritage Dictionary:

sum·ma·tion

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(sə-mā'shən) pronunciation

The act or process of adding; addition.
A sum or aggregate.
A concluding part of a speech or argument containing a summary of principal points, especially of a case before a court of law.
Physiology. The process by which multiple or repeated stimuli can produce a response in a nerve, muscle, or other part that one stimulus alone cannot produce.

[New Latin summātiō, summātiōn-, from Late Latin summātus, past participle of summāre, to sum up, from Latin summa, sum. See sum.]

summation

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noun

The act or process of adding: addition, totalization. See increase/decrease.
A number or quantity obtained as a result of addition: aggregate, amount, sum, sum total, total, totality. Archaic tale. See count.
A condensation of the essential or main points of something: recapitulation, rundown, run-through, sum, summary, summing-up, wrap-up. Informal recap. See words.

Oxford Dictionary of Sports Science & Medicine:

summation

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The summing of all the individual changes elicited by different stimuli on the membrane potential of a neurone or muscle fibre. Summation may involve two or more excitatory stimuli, inhibitory stimuli, or a combination of excitatory and inhibitory stimuli.

Barron's Law Dictionary:

summation

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The final step in a trial , wherein each party’s counsel reviews the evidence that has been presented and attempts to show why its position should prevail; also known as closing arguments.
In a jury trial, this step immediately precedes a judge’s instructions to a jury. The party with the burden of proof always closes or sums up last.
Therefore, in a civil case, the defendant closes first and then the plaintiff follows.
In criminal cases, the procedure varies among jurisdictions. Fed. R.
Crim. P. 29.1 provides that the prosecution closes first, with the defendant following.
In most cases, the prosecution is also afforded an opportunity to rebut the defendant’s closing as well.
A prosecutor has the special burden to prove the state’s allegations beyond a reasonable doubt. During summation a prosecutor must not: comment on a defendant’s failure to testify, 380 U.S.
609; refer to evidence not in record, 595 F. 2d 751; interject personal opinions concerning veracity of witnesses, 543 F. 2d 1333; appeal to a jury based on passion or prejudice rather than facts, 664 F. 2d 971; or imply that the prosecutor believes that the defendant is guilty of the crime charged, 558 F.
2d 387. A prosecutor cannot be argumentative in his or her closing and “it is as much [the prosecutor’s] duty to refrain from improper methods calculated to produce a wrongful conviction as it is to use every legitimate means to bring about a just one.” 295 U.S.
78, 88. Failure of the prosecutor to comply with the above is referred to as prosecutorial misconduct and may result in a mistrial. Still, the prosecutor is entitled to a certain degree of latitude in summation, 553 F. 2d 1013, and his or her closing must be viewed in the context of the entire trial rather than in the abstract. 666 F. 2d 1227.

Oxford Dictionary of Biochemistry:

summation

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the determination of the sum (def. 1) of a series.

Previous:	sumatriptan, sum, sulph
Next:	super+, super-secondary structure, superantigen

Saunders Veterinary Dictionary:

summation

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Accumulate, add up, aggregate a series of numbers or quantities or events.

s. of effects — a theory explaining clinical pruritus as the additive effects of pruritus from several causes which may raise the individual above the threshold, but pruritus from any single cause would be unlikely to do so.
s. gallop — see gallop rhythm.
neurological s. — physiological summation in synapses is a characteristic of the mammalian nervous system. It may be spatial, with additional synaptic junctions participating, or temporal, when succeeding stimuli catch up with the as-yet undischarged neurotransmitter. Seen in the retina of the cat, as an example of a nocturnal animal, where many millions of photoreceptors are connected to only one million axons, resulting in maximal sensitivity to light.
weighted s. — the sum obtained by adding the numerical value for individual clinical signs, each weighted to express their importance, when making a diagnosis. The total, as a fraction or a percentage, provides an estimate of the probability of each diagnosis being the correct one.

Mosby's Dental Dictionary:

summation

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The phenomenon in which similar actions of more than one drug result in a total action that may be expressed as the arithmetical sum of the effects of the individual drugs.

Random House Word Menu:

categories related to 'summation'

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For a list of words related to summation, see:

Physiology - summation: temporal or spatial addition of electrical effects of multiple stimuli
Customs, Formalities, and Practices - summation: lawyer’s final argument and closing statement to jury
Legal Procedures and Trials - summation: attorney’s concluding remarks to jury

Wikipedia on Answers.com:

Summation

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"Sum" and "Summation" redirect here. For other uses, see Sum (disambiguation) and Summation (disambiguation).

Summation is the operation of 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 integers, rational numbers, real numbers, or complex numbers. Besides numbers, other types of values can be added as well: vectors, matrices, polynomials and, in general, elements of any 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 sums).

Summation of an infinite sequence of values is not always possible, and when a value can be given for an infinite summation, this involves more than just the addition operation, namely also the notion of a limit. Such infinite summations are known as series. Another notion involving limits of finite sums is integration. The term summation has a special meaning related to extrapolation in the context of divergent series.

The summation of the sequence [1, 2, 4, 2] is an expression whose value is the sum of each of the members of the sequence. In the example,1 + 2 + 4 + 2 = 9. Since addition is associative the value does not depend on how the additions are grouped, for instance (1 + 2) + (4 + 2) and 1 + ((2 + 4) + 2) both have the value 9; therefore, parentheses are usually omitted in repeated additions. Addition is also commutative, so 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).

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: 1 + 2 + 3 + ... + 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 "Σ". Using this notation the above summation is written as:

$\sum_{i=1}^{100}i.$

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

$\sum_{i=1}^ni = \frac{n(n+1)}2$

for all natural numbers n. More generally, formulas exist for many summations of terms following a regular pattern.

The term "indefinite summation" 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 differences are given by s. By contrast, summation as discussed in this article is called "definite summation".

Contents

1 Notation
- 1.1 Capital-sigma notation
- 1.2 Special cases
2 Formal Definition
3 Measure theory notation
4 Fundamental theorem of discrete calculus
5 Approximation by definite integrals
6 Identities
7 Growth rates
8 See also
9 Notes
10 Further reading
11 External links

Notation

Capital-sigma notation

Mathematical notation uses a symbol that compactly represents summation of many similar terms: the summation symbol, ∑, an enlarged form of the upright capital Greek letter Sigma. This is defined as:

$\sum_{i=m}^n x_i = x_m + x_{m+1} + x_{m+2} +\cdots+ x_{n-1} + x_n.$

Where, i represents the index of summation; x_i 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.

Here is an example showing the summation of exponential terms (all terms to the power of 2):

$\sum_{i=3}^6 i^2 = 3^2+4^2+5^2+6^2 = 86.$

Informal writing sometimes omits the definition of the index and bounds of summation when these are clear from context, as in:

$\sum x_i^2 = \sum_{i=1}^n x_i^2.$

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:

$\sum_{0\le k< 100} f(k)$

is the sum of f(k) over all (integer) k in the specified range,

$\sum_{x\in S} f(x)$

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

$\sum_{d|n}\;\mu(d)$

is the sum of μ(d) over all positive integers d dividing n.^[1]

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

$\sum_{\ell,\ell'}$

is the same as

$\sum_\ell\sum_{\ell'}.$

A similar notation is applied when it comes to denoting the 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, replacing the ∑.

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 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 m = n in the definition above, then there is only one term in the sum; if m > n, then there is none.

Formal Definition

If the iterated function notation is defined e.g. $f^2(x) \equiv f(f(x))$ and is considered a more primitive notation then summation can be defined in terms of iterated functions as:

$\left\{b+1,\sum_{i=a}^b g(i)\right\} \equiv \left( \{i,x\} \rightarrow \{ i+1 ,x+g(i) \}\right)^{b-a+1} \{a,0\}$

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}.

Measure theory notation

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

$\sum_{k=a}^b f(k) = \int_{[a,b]} f\,d\mu$

where $[a, b]$ is the subset of the integers from $a$ to $b$ , and where $μ$ is the counting measure.

Fundamental theorem of discrete calculus

Indefinite sums can be used to calculate definite sums with the formula^[2]:

$\sum_{k=a}^b f(k)=\Delta^{-1}f(b+1)-\Delta^{-1}f(a)$

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:

$\int_{s=a-1}^{b} f(s)\ ds \le \sum_{i=a}^{b} f(i) \le \int_{s=a}^{b+1} f(s)\ ds.$

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 formulas below involve finite sums; for infinite summations see list of mathematical series

General manipulations

$\sum_{n=s}^t C\sdot f(n) = C\sdot \sum_{n=s}^t f(n)$ , where C is a constant

$\sum_{n=s}^t f(n) + \sum_{n=s}^{t} g(n) = \sum_{n=s}^t \left[f(n) + g(n)\right]$

$\sum_{n=s}^t f(n) - \sum_{n=s}^{t} g(n) = \sum_{n=s}^t \left[f(n) - g(n)\right]$

$\sum_{n=s}^t f(n) = \sum_{n=s+p}^{t+p} f(n-p)$

$\sum_{n=s}^j f(n) + \sum_{n=j+1}^t f(n) = \sum_{n=s}^t f(n)$

$\left(\sum_{i=k_0}^{k_1} a_i\right)\left(\sum_{j=l_0}^{l_1} b_j\right) = \sum_{i=k_0}^{k_1}\sum_{j=l_0}^{l_1} a_ib_j$

$\sum_{i=k_0}^{k_1}\sum_{j=l_0}^{l_1} a_{i,j} = \sum_{j=l_0}^{l_1}\sum_{i=k_0}^{k_1} a_{i,j}$

$\sum_{n=0}^t f(2n) + \sum_{n=0}^t f(2n+1) = \sum_{n=0}^{2t+1} f(n)$

$\sum_{n=0}^t \sum_{i=0}^{z-1} f(z\sdot n+i) = \sum_{n=0}^{z\sdot t+z-1} f(n)$

$\sum_{n=s}^t \ln f(n) = \ln \prod_{n=s}^t f(n)$

$c^{\left[\sum_{n=s}^t f(n) \right]} = \prod_{n=s}^t c^{f(n)}$

Some summations of polynomial expressions

$\sum_{i=m}^n 1 = n+1-m$

$\sum_{i=1}^n \frac{1}{i} = H_n$ (See Harmonic number)

$\sum_{i=m}^n i = \frac{(n+1-m)(n+m)}{2}$ (see arithmetic series)

$\sum_{i=0}^n i = \sum_{i=1}^n i = \frac{n(n+1)}{2}$ (Special case of the arithmetic series)

$\sum_{i=0}^n i^2 = \frac{n(n+1)(2n+1)}{6} = \frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6}$

$\sum_{i=0}^n i^3 = \left(\frac{n(n+1)}{2}\right)^2 = \frac{n^4}{4} + \frac{n^3}{2} + \frac{n^2}{4} = \left[\sum_{i=1}^n i\right]^2$

$\sum_{i=0}^n i^4 = \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30} = \frac{n^5}{5} + \frac{n^4}{2} + \frac{n^3}{3} - \frac{n}{30}$

$\sum_{i=0}^n i^p = \frac{(n+1)^{p+1}}{p+1} + \sum_{k=1}^p\frac{B_k}{p-k+1}{p\choose k}(n+1)^{p-k+1}$ where $B_k$ denotes a Bernoulli number

The following formulas are manipulations of $\sum_{i=0}^n i^3 = \left(\sum_{i=0}^n i\right)^2$ generalized to begin a series at any natural number value (i.e., $m \in \mathbb{N}$ ):

$\left(\sum_{i=m}^n i\right)^2 = \sum_{i=m}^n ( i^3 - im(m-1) )$

$\sum_{i=m}^n i^3 = \left(\sum_{i=m}^n i\right)^2 +\sum_{i=m}^n i$

Some summations involving exponential terms

In the summations below x is a constant not equal to 1

$\sum_{i=m}^{n-1} x^i = \frac{x^m-x^n}{1-x}$ (m < n; see geometric series)

$\sum_{i=0}^{n-1} x^i = \frac{1-x^n}{1-x}$ (geometric series starting at 1)

$\sum_{i=0}^{n-1} i x^i = \frac{x-nx^n+(n-1)x^{n+1}}{(1-x)^2}$

$\sum_{i=0}^{n-1} i 2^i = 2+(n-2)2^{n}$ (special case when x = 2)

$\sum_{i=0}^{n-1} \frac{i}{2^i} = 2-\frac{n+1}{2^{n-1}}$ (special case when x = 1/2)

Some summations involving binomial coefficients

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.

$\sum_{i=0}^n {n \choose i} = 2^n$

$\sum_{i=1}^{n} i{n \choose i} = n2^{n-1}$

$\sum_{i=0}^{n} i!\cdot{n \choose i} = \lfloor n!\cdot e \rfloor$

$\sum_{i=0}^{n-1} {i \choose k} = {n \choose k+1}$

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

Growth rates

The following are useful approximations (using theta notation):

$\sum_{i=1}^n i^c = \Theta(n^{c+1})$ for real c greater than −1

$\sum_{i=1}^n \frac{1}{i} = \Theta(\log n)$ (See Harmonic number)

$\sum_{i=1}^n c^i = \Theta(c^n)$ for real c greater than 1

$\sum_{i=1}^n \log(i)^c = \Theta(n \cdot \log(n)^{c})$ for non-negative real c

$\sum_{i=1}^n \log(i)^c \cdot i^d = \Theta(n^{d+1} \cdot \log(n)^{c})$ for non-negative real c, d

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

Notes

^ Although the name of the 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.
^ "Handbook of discrete and combinatorial mathematics", Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, ISBN 0-8493-0149-1

External links

Media related to Summation at Wikimedia Commons
Summation, PlanetMath.org.
Derivation of Polynomials to Express the Sum of Natural Numbers with Exponents

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Translations:

Summation

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Dansk (Danish)
n. - sammenlægning, summering

Nederlands (Dutch)
sommatie, opsomming

Français (French)
n. - résumé, récapitulation, total, (US, Jur) conclusions

Deutsch (German)
n. - Addition

Ελληνική (Greek)
n. - άθροιση, πρόσθεση, ανακεφαλαίωση

Italiano (Italian)
somma, riassunto, arringa finale

Português (Portuguese)
n. - sumário (m)

Русский (Russian)
сложение, суммирование, совокупность, итог, заключительная речь судьи, резюме, оценка (положения), подведение итогов

Español (Spanish)
n. - suma, adición, total, recapitulación, resumen

Svenska (Swedish)
n. - summering, sammanräkning, sammanfattning, resumé, slutsumma

中文（简体）(Chinese (Simplified))
总和, 合计, 和

中文（繁體）(Chinese (Traditional))
n. - 總和, 合計, 和

한국어 (Korean)
n. - 합계하기, 간추리기, 최종 변론

日本語 (Japanese)
n. - 加法, 合計, 要約, 最終弁論, 加重

العربيه (Arabic)
‏(الاسم) اضافه , جمع‏

עברית (Hebrew)
n. - ‮חיבור, סיכום, תמצית‬

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American Heritage Dictionary The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read

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Roget's Thesaurus Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 byHoughton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved. Read

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summation

sum·ma·tion

summation

summation

summation

summation

summation

summation

categories related to 'summation'

Summation

Notation

Capital-sigma notation

Special cases

Formal Definition

Measure theory notation

Fundamental theorem of discrete calculus

Approximation by definite integrals

Identities

General manipulations

Some summations of polynomial expressions

Some summations involving exponential terms

Some summations involving binomial coefficients

Growth rates

See also

Notes

Further reading

External links

Summation

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