TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, summation is the
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

of a
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of any kind of
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

s, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well:
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
,
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
,
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
,
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s and, in general, elements of any type of
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs ...
s on which an operation denoted "+" is defined. Summations of
infinite sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s are called
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
. They involve the concept of
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
, and are not considered in this article. The summation of an explicit sequence is denoted as a succession of additions. For example, summation of is denoted , and results in 9, that is, . Because addition is
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
and
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, there is no need of 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 regular pattern, as a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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 File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
s may be written as . Otherwise, summation is denoted by using Σ notation, where $\sum$ is an enlarged capital
Greek letter The Greek alphabet has been used to write the Greek language Greek (modern , romanized: ''Elliniká'', Ancient Greek, ancient , ''Hellēnikḗ'') is an independent branch of the Indo-European languages, Indo-European family of languages, nat ...
sigma Sigma (uppercase Letter case is the distinction between the Letter (alphabet), letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the written represent ...

. For example, the sum of the first natural numbers can be denoted as $\sum_^n i.$ For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find
closed-form expression In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s for the result. For example, :$\sum_^n i = \frac.$ 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

Capital-sigma notation

Mathematical notation uses a symbol that compactly represents summation of many similar terms: the ''summation symbol'', $\sum$, an enlarged form of the upright capital Greek letter
sigma Sigma (uppercase Letter case is the distinction between the Letter (alphabet), letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the written represent ...

. This is defined as :$\sum_^n a_i = a_m + a_ + a_ + \cdots + a_ + a_n$ where is the index of summation; is an indexed variable representing each term of the sum; is the lower bound of summation, and is the upper bound of summation. The "" under the summation symbol means that the index starts out equal to . The index, , is incremented by one for each successive term, stopping when . This is read as "sum of , from to ". Here is an example showing the summation of squares: :$\sum_^6 i^2 = 3^2+4^2+5^2+6^2 = 86.$ 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 to ''n''. For example, one might write that: :$\sum a_i^2 = \sum_^n a_i^2.$ 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: :$\sum_ f\left(k\right)$ is an alternative notation for $\sum_^ f(k),$ the sum of $f\left(k\right)$ over all (
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
s) $k$ in the specified range. Similarly, :$\sum_ f\left(x\right)$ is the sum of $f\left(x\right)$ over all elements $x$ in the set $S$, and :$\sum_\;\mu\left(d\right)$ is the sum of $\mu\left(d\right)$ over all positive integers $d$ $n$. There are also ways to generalize the use of many sigma signs. For example, :$\sum_$ is the same as :$\sum_\sum_.$ A similar notation is used for the
product of a sequence Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on Computer, computers, by an asterisk ) is one of the four Elementary arithmetic, eleme ...
, where $\prod$, an enlarged form of the Greek capital letter pi, is used instead of $\sum.$

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 is
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

, because zero is the
identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
for addition. This is known as the ''
empty sum In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
''. 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.

Formal definition

Summation may be defined recursively as follows: :$\sum_^b g\left(i\right)=0$, for ''b'' < ''a''; : :$\sum_^b g\left(i\right)=g\left(b\right)+\sum_^ g\left(i\right)$, for ''b'' ≥ ''a''.

Measure theory notation

In the notation of measure and theory, a sum can be expressed as a
definite integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, :$\sum_^b f\left(k\right) = \int_ f\,d\mu$ where 
, b The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
/math> is the
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of the integers from $a$ to $b$, and where $\mu$ is the
counting measureIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
.

Calculus of finite differences

Given a function that is defined over the integers in the interval , the following equation holds: :$f\left(n\right)-f\left(m\right)= \sum_^ \left(f\left(i+1\right)-f\left(i\right)\right).$ This is the analogue of the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating the gradient) with the concept of integral, integrating a function (calculating the area under t ...
in calculus of finite differences, which states that: :$f\left(n\right)-f\left(m\right)=\int_m^n f\text{'}\left(x\right)\,dx,$ where :$f\text{'}\left(x\right)=\lim_ \frac$ is the
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of . An example of application of the above equation is the following: :$n^k=\sum_^ \left\left(\left(i+1\right)^k-i^k\right\right).$ Using
binomial theorem In elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...
, this may be rewritten as: :$n^k=\sum_^ \left\left(\sum_^ \binom i^j\right\right).$ The above formula is more commonly used for inverting of the
difference operator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
$\Delta$, defined by: :$\Delta\left(f\right)\left(n\right)=f\left(n+1\right)-f\left(n\right),$ where is a function defined on the nonnegative integers. Thus, given such a function , the problem is to compute the antidifference of , a function $F=\Delta^f$ such that $\Delta F=f$. That is, $F\left(n+1\right)-F\left(n\right)=f\left(n\right).$ This function is defined up to the addition of a constant, and may be chosen as''Handbook of Discrete and Combinatorial Mathematics'', Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, . :$F\left(n\right)=\sum_^ f\left(i\right).$ There is not always a
closed-form expression In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
for such a summation, but
Faulhaber's formula In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
provides a closed form in the case where $f\left(n\right)=n^k$ and, by
linearity Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to ''Proportionality (math ...

, for every
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
of .

Approximation by definite integrals

Many such approximations can be obtained by the following connection between sums and
integral In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s, which holds for any
increasing Image:Monotonicity example3.png, Figure 3. A function that is not monotonic In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that pres ...
function ''f'': :$\int_^ f\left(s\right)\ ds \le \sum_^ f\left(i\right) \le \int_^ f\left(s\right)\ ds.$ and for any decreasing function ''f'': :$\int_^ f\left(s\right)\ ds \le \sum_^ f\left(i\right) \le \int_^ f\left(s\right)\ ds.$ For more general approximations, see the
Euler–Maclaurin formula In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related Summation, sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and series (mathe ...
. For summations in which the summand is given (or can be interpolated) by an
integrableIn mathematics, integrability is a property of certain dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometric ...
function of the index, the summation can be interpreted as a
Riemann sum In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
occurring in the definition of the corresponding definite integral. One can therefore expect that for instance :$\frac\sum_^ f\left\left(a+i\fracn\right\right) \approx \int_a^b f\left(x\right)\ dx,$ 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 formulae below involve finite sums; for infinite summations or finite summations of expressions involving
trigonometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s or other
transcendental function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, see
list of mathematical series This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. *Here, 0^0 Zero to the power of zero , is taken to have the value 1 *B_n(x) is a Bernoulli polynom ...
.

General identities

: $\sum_^t C\cdot f\left(n\right) = C\cdot \sum_^t f\left(n\right) \quad$ (
distributivity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
) : $\sum_^t f\left(n\right) \pm \sum_^ g\left(n\right) = \sum_^t \left\left(f\left(n\right) \pm g\left(n\right)\right\right)\quad$ (
commutativity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

and
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...
) : $\sum_^t f\left(n\right) = \sum_^ f\left(n-p\right)\quad$ (index shift) : $\sum_ f\left(n\right) = \sum_ f\left(\sigma\left(m\right)\right), \quad$ for a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

from a finite set onto a set (index change); this generalizes the preceding formula. : $\sum_^t f\left(n\right) =\sum_^j f\left(n\right) + \sum_^t f\left(n\right)\quad$ (splitting a sum, using
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...
) : $\sum_^f\left(n\right)=\sum_^f\left(n\right)-\sum_^f\left(n\right)\quad$ (a variant of the preceding formula) : $\sum_^t f\left(n\right) = \sum_^ f\left(t-n\right)\quad$ (the sum from the first term up to the last is equal to the sum from the last down to the first) : $\sum_^t f\left(n\right) = \sum_^ f\left(t-n\right)\quad$ (a particular case of the formula above) : $\sum_^\sum_^ a_ = \sum_^\sum_^ a_\quad$ (commutativity and associativity, again) : $\sum_ a_ = \sum_^n\sum_^i a_ = \sum_^n\sum_^n a_ = \sum_^\sum_^ a_\quad$ (another application of commutativity and associativity) : $\sum_^ f\left(n\right) = \sum_^t f\left(2n\right) + \sum_^t f\left(2n+1\right)\quad$ (splitting a sum into its odd and even parts, for even indexes) : $\sum_^ f\left(n\right) = \sum_^t f\left(2n\right) + \sum_^t f\left(2n-1\right)\quad$ (splitting a sum into its odd and even parts, for odd indexes) :$\left\left(\sum_^ a_i\right\right) \left\left(\sum_^ b_j\right\right)=\sum_^n \sum_^n a_ib_j \quad$ (
distributivity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
) : $\sum_^m\sum_^n = \left\left(\sum_^m a_i\right\right) \left\left( \sum_^n c_j \right\right)\quad$ (distributivity allows factorization) : $\sum_^t \log_b f\left(n\right) = \log_b \prod_^t f\left(n\right)\quad$ (the
logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

of a product is the sum of the logarithms of the factors) : $C^ = \prod_^t C^\quad$ (the
exponential Exponential may refer to any of several mathematical topics related to exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " raise ...
of a sum is the product of the exponential of the summands)

Powers and logarithm of arithmetic progressions

: $\sum_^n c = nc\quad$ for every that does not depend on : $\sum_^n i = \sum_^n i = \frac\qquad$ (Sum of the simplest
arithmetic progression An Arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common diffe ...

, consisting of the first ''n'' natural numbers.) : $\sum_^n \left(2i-1\right) = n^2\qquad$ (Sum of first odd natural numbers) : $\sum_^ 2i = n\left(n+1\right)\qquad$ (Sum of first even natural numbers) : $\sum_^ \log i = \log n!\qquad$ (A sum of
logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

s is the logarithm of the product) : $\sum_^n i^2 = \sum_^n i^2 = \frac = \frac + \frac + \frac\qquad$ (Sum of the first
squares In geometry, a square is a regular polygon, regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree (angle), degree angles, or 100-gradian angles or right angles). It can also be defined as a rectangle in ...
, see
square pyramidal number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.) : $\sum_^n i^3 = \left\left(\sum_^n i \right\right)^2 = \left\left(\frac\right\right)^2 = \frac + \frac + \frac\qquad$ (
Nicomachus's theorem In number theory, the sum of the first Cube (algebra), cubes is the Square number, square of the th triangular number. That is, :1^3+2^3+3^3+\cdots+n^3 = \left(1+2+3+\cdots+n\right)^2. The same equation may be written more compactly using the m ...
) More generally, one has
Faulhaber's formula In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
for $p>1$ : $\sum_^n k^ = \frac + \fracn^p + \sum_^p \binom p k \frac\,n^,$ where $B_k$ denotes a
Bernoulli number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, and $\binom p k$ is a
binomial coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
.

Summation index in exponents

In the following summations, is assumed to be different from 1. : $\sum_^ a^i = \frac$ (sum of a
geometric progression In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

) : $\sum_^ \frac = 2-\frac$ (special case for ) : $\sum_^ i a^i =\frac$ ( times the derivative with respect to of the geometric progression) : $\begin \sum_^ \left\left(b + i d\right\right) a^i &= b \sum_^ a^i + d \sum_^ i a^i\\ & = b \left\left(\frac\right\right) + d \left\left(\frac\right\right)\\ & = \frac+\frac \end$ :::(sum of an arithmetico–geometric sequence)

Binomial coefficients and factorials

There exist very many summation identities involving binomial coefficients (a whole chapter of ''
Concrete Mathematics ''Concrete Mathematics: A Foundation for Computer Science'', by Ronald Graham, Donald Knuth, and Oren Patashnik, first published in 1989, is a textbook that is widely used in computer-science departments as a substantive but light-hearted treatm ...
'' is devoted to just the basic techniques). Some of the most basic ones are the following.

Involving the binomial theorem

: $\sum_^n a^ b^i=\left(a + b\right)^n,$ the
binomial theorem In elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...
: $\sum_^n = 2^n,$ the special case where : $\sum_^n p^i \left(1-p\right)^=1$, the special case where , which, for $0 \le p \le 1,$ expresses the sum of the
binomial distribution In probability theory and statistics, the Binomial coefficient, binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' statistical independence, indep ...

: $\sum_^ i = n\left(2^\right),$ the value at of the
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

with respect to of the binomial theorem : $\sum_^n \frac = \frac,$ the value at of the
antiderivative In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zer ...
with respect to of the binomial theorem

Involving permutation numbers

In the following summations, $_P_$ is the number of -permutations of . : $\sum_^ _P_ = _P_\left(2^\right)$ : $\sum_^n _P_ = \sum_^n \prod_^k \left(i+j\right) = \frac$ : $\sum_^ i!\cdot = \sum_^ _P_ = \lfloor n! \cdot e \rfloor, \quad n \in \mathbb^+$, where and $\lfloor x\rfloor$ denotes the
floor function In mathematics and computer science, the floor function is the function (mathematics), function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function ...

.

Others

: $\sum_^ \binom = \binom$ : $\sum_^ =$ : $\sum_^n i\cdot i! = \left(n+1\right)! - 1$ : $\sum_^n =$ :$\sum_^n ^2 =$ :$\sum_^n \frac = \frac$

Harmonic numbers

: $\sum_^n \frac = H_n$ (that is the th
harmonic number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
) : $\sum_^n \frac = H^k_n$ (that is a generalized harmonic number)

Growth rates

The following are useful
approximation An approximation is anything that is intentionally similar but not exactly equal Equal or equals may refer to: Arts and entertainment * Equals (film), ''Equals'' (film), a 2015 American science fiction film * Equals (game), ''Equals'' (game), a ...
s (using theta notation): : $\sum_^n i^c \in \Theta\left(n^\right)$ for real ''c'' greater than −1 : : $\sum_^n \frac \in \Theta\left(\log_e n\right)$ (See
Harmonic number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
) : : $\sum_^n c^i \in \Theta\left(c^n\right)$ for real ''c'' greater than 1 : : $\sum_^n \log\left(i\right)^c \in \Theta\left(n \cdot \log\left(n\right)^\right)$ for
non-negative In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third ...
real ''c'' : : $\sum_^n \log\left(i\right)^c \cdot i^d \in \Theta\left(n^ \cdot \log\left(n\right)^\right)$ for non-negative real ''c'', ''d'' : : $\sum_^n \log\left(i\right)^c \cdot i^d \cdot b^i \in \Theta \left(n^d \cdot \log\left(n\right)^c \cdot b^n\right)$ for non-negative real ''b'' > 1, ''c'', ''d''

* Capital-pi notation *
Einstein notation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
*
Iverson bracketIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
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Iterated binary operationIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
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Kahan summation algorithmIn numerical analysis, the Kahan summation algorithm, also known as compensated summation, significantly reduces the numerical error in the total obtained by adding a sequence of finite-decimal precision, precision floating-point numbers, compared to ...
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Product (mathematics) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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Summation by parts In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
* the summation single glyph (U+2211 ''N-ARY SUMMATION'') * the paired glyph's beginning (U+23B2 ''SUMMATION TOP'') * the paired glyph's end (U+23B3 ''SUMMATION BOTTOM'')