In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
Summations of infinite sequences are called series. They involve the concept of 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 is denoted, and results in 9, that is, . 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 element results in this element itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0.
Very often, the elements of a sequence are defined, through a regular pattern, as a 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 numbers may be written as . Otherwise, summation is denoted by using Σ notation, where is an enlarged capital Greek letter sigma. For example, the sum of the first natural numbers can be denoted as
For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find closed-form expressions for the result. For example,
n | |
\sum | |
i=1 |
i=
n(n+1) | |
2 |
.
Although such formulas do not always exist, many summation formulas have been discovered—with some of the most common and elementary ones being listed in the remainder of this article.
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
n | |
\sum | |
i=m |
ai=am+am+1+am+2+ … +an-1+an
This is read as "sum of, from to ".
Here is an example showing the summation of squares:
6 | |
\sum | |
i=3 |
i2=32+42+52+62=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
n
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.[1] For example, one might write that:
\sum
2 | |
a | |
i |
=
n | |
\sum | |
i=1 |
2. | |
a | |
i |
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:
\sum0f(k)
f(k)
k
\sumxf(x)
f(x)
x
S
\sumd|n \mu(d)
\mu(d)
d
n
There are also ways to generalize the use of many sigma signs. For example,
\sumi,j
\sumi\sumj.
A similar notation is used for the product of a sequence, where , an enlarged form of the Greek capital letter pi, is used instead of
It is possible to sum fewer than 2 numbers:
x
x
These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case.For example, if
n=m
n=m-1
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.
Summation may be defined recursively as follows:
b | |
\sum | |
i=a |
g(i)=0
b<a
b | |
\sum | |
i=a |
b-1 | |
g(i)=g(b)+\sum | |
i=a |
g(i)
b\geqslanta
In the notation of measure and integration theory, a sum can be expressed as a definite integral,
b | |
\sum | |
k=a |
f(k)=\int[a,b]fd\mu
where
[a,b]
a
b
\mu
Given a function that is defined over the integers in the interval, the following equation holds:
f(n)-f(m)=
n-1 | |
\sum | |
i=m |
(f(i+1)-f(i)).
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:
n | |
f(n)-f(m)=\int | |
m |
f'(x)dx,
f'(x)=\limh\to
f(x+h)-f(x) | |
h |
An example of application of the above equation is the following:
n-1 | |
n | |
i=0 |
\left((i+1)k-ik\right).
n-1 | |
n | |
i=0 |
k-1 | |
l(\sum | |
j=0 |
\binom{k}{j}ijr).
\Delta
\Delta(f)(n)=f(n+1)-f(n),
F=\Delta-1f
\DeltaF=f
F(n+1)-F(n)=f(n).
n-1 | |
F(n)=\sum | |
i=0 |
f(i).
There is not always a closed-form expression for such a summation, but Faulhaber's formula provides a closed form in the case where
f(n)=nk
Many such approximations can be obtained by the following connection between sums and integrals, which holds for any increasing function f:
b | |
\int | |
s=a-1 |
f(s) ds\le
b | |
\sum | |
i=a |
f(i)\le
b+1 | |
\int | |
s=a |
f(s) ds.
and for any decreasing function f:
b+1 | |
\int | |
s=a |
f(s) ds\le
b | |
\sum | |
i=a |
f(i)\le
b | |
\int | |
s=a-1 |
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
b-a | |
n |
n-1 | ||
\sum | f\left(a+i | |
i=0 |
b-a | |
n\right) |
≈
b | |
\int | |
a |
f(x) dx,
since the right-hand side is by definition the limit for
n\toinfty
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.
t | |
\sum | |
n=s |
C ⋅ f(n)=C ⋅
t | |
\sum | |
n=s |
f(n)
t | |
\sum | |
n=s |
f(n)\pm
t | |
\sum | |
n=s |
g(n)=
t | |
\sum | |
n=s |
\left(f(n)\pmg(n)\right)
t | |
\sum | |
n=s |
f(n)=
t+p | |
\sum | |
n=s+p |
f(n-p)
\sumn\inf(n)=\summ\inf(\sigma(m)),
t | |
\sum | |
n=s |
f(n)
j | |
=\sum | |
n=s |
f(n)+
t | |
\sum | |
n=j+1 |
f(n)
b | |
\sum | |
n=a |
b | |
f(n)=\sum | |
n=0 |
a-1 | |
f(n)-\sum | |
n=0 |
f(n)
t | |
\sum | |
n=s |
f(n)=
t-s | |
\sum | |
n=0 |
f(t-n)
t | |
\sum | |
n=0 |
f(n)=
t | |
\sum | |
n=0 |
f(t-n)
k1 | |
\sum | |
i=k0 |
l1 | |
\sum | |
j=l0 |
ai,j=
l1 | |
\sum | |
j=l0 |
k1 | |
\sum | |
i=k0 |
ai,j
\sumk\leai,j=
i | |
\sum | |
j=k |
ai,j=
n | |
\sum | |
i=j |
ai,j
n-k | |
= \sum | |
j=0 |
n-j | |
\sum | |
i=k |
ai+j,i
2t+1 | |
\sum | |
n=2s |
f(n)=
t | |
\sum | |
n=s |
f(2n)+
t | |
\sum | |
n=s |
f(2n+1)
2t | |
\sum | |
n=2s+1 |
f(n)=
t | |
\sum | |
n=s+1 |
f(2n)+
t | |
\sum | |
n=s+1 |
f(2n-1)
n | |
l(\sum | |
i=0 |
air)
n | |
l(\sum | |
j=0 |
bjr)=\sum
n | |
i=0 |
n | |
\sum | |
j=0 |
aibj
n | |
\sum | |
j=t |
{ai}{cj}=
m | |
l(\sum | |
i=s |
air)l(
n | |
\sum | |
j=t |
cjr)
t | |
\sum | |
n=s |
logbf(n)=logb
t | |
\prod | |
n=s |
f(n)
| ||||||||||
C |
=
t | |
\prod | |
n=s |
Cf(n)
k | |
\sum | |
m=0 |
m | |
\sum | |
n=0 |
k | |
f(m,n)=\sum | |
m=0 |
k | |
\sum | |
n=m |
f(n,m),
n | |
\sum | |
i=1 |
c=nc
n | |
\sum | |
i=0 |
i=
n | |
\sum | |
i=1 |
i=
n(n+1) | |
2 |
n | |
\sum | |
i=1 |
(2i-1)=n2
n | |
\sum | |
i=0 |
2i=n(n+1)
n | |
\sum | |
i=1 |
logi=logn!
n | |
\sum | |
i=0 |
i2=
n | |
\sum | |
i=1 |
i2=
n(n+1)(2n+1) | |
6 |
=
n3 | |
3 |
+
n2 | |
2 |
+
n | |
6 |
n | |
\sum | |
i=0 |
i3=
n | |
l(\sum | |
i=0 |
ir)2=\left(
n(n+1) | |
2 |
\right)2=
n4 | |
4 |
+
n3 | |
2 |
+
n2 | |
4 |
More generally, one has Faulhaber's formula for
p>1
n | |
\sum | |
k=1 |
kp=
np+1 | |
p+1 |
+
1 | |
2 |
np+
p | |
\sum | |
k=2 |
\binompk
Bk | |
p-k+1 |
np-k+1,
Bk
\binompk
In the following summations, is assumed to be different from 1.
n-1 | |
\sum | |
i=0 |
ai=
1-an | |
1-a |
n-1 | |
\sum | |
i=0 |
1 | |
2i |
=2-
1 | |
2n-1 |
n-1 | |
\sum | |
i=0 |
iai=
a-nan+(n-1)an+1 | |
(1-a)2 |
\begin
n-1 | |
{align} \sum | |
i=0 |
\left(b+id\right)ai&=b
n-1 | |
\sum | |
i=0 |
ai+d
n-1 | |
\sum | |
i=0 |
iai\\ &=b\left(
1-an | |
1-a |
\right)+d\left(
a-nan+(n-1)an+1 | |
(1-a)2 |
\right)\\ &=
b(1-an)-(n-1)dan | + | |
1-a |
da(1-an) | |
(1-a)2 |
\end{align}
(sum of an arithmetico–geometric sequence)
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.
n | |
\sum | |
i=0 |
{n\choosei}an-ibi=(a+b)n,
n | |
\sum | |
i=0 |
{n\choosei}=2n,
n | |
\sum | |
i=0 |
{n\choosei}pi(1-p)n-i=1
0\lep\le1,
n | |
\sum | |
i=0 |
i{n\choosei}=n(2n-1),
n | |
\sum | |
i=0 |
n\choosei | |
i+1 |
=
2n+1-1 | |
n+1 |
,
In the following summations,
{}nPk
n | |
\sum | |
i=0 |
{}iPk{n\choosei}={}nPk(2n-k)
n | |
\sum | |
i=1 |
{}i+kPk+1=
n | |
\sum | |
i=1 |
k | |
\prod | |
j=0 |
(i+j)=
(n+k+1)! | |
(n-1)!(k+2) |
n | |
\sum | |
i=0 |
i! ⋅ {n\choosei}=
n | |
\sum | |
i=0 |
{}nPi=\lfloorn! ⋅ e\rfloor, n\inZ+
\lfloorx\rfloor
m | |
\sum | |
k=0 |
\binom{n+k}{n}=\binom{n+m+1}{n+1}
n | |
\sum | |
i=k |
{i\choosek}={n+1\choosek+1}
n | |
\sum | |
i=0 |
i ⋅ i!=(n+1)!-1
n | |
\sum | |
i=0 |
{m+i-1\choosei}={m+n\choosen}
n | |
\sum | |
i=0 |
{n\choosei}2={2n\choosen}
n | |
\sum | |
i=0 |
1 | |
i! |
=
\lfloorn! e\rfloor | |
n! |
n | |
\sum | |
i=1 |
1 | |
i |
=Hn
n | |
\sum | |
i=1 |
1 | |
ik |
=
k | |
H | |
n |
The following are useful approximations (using theta notation):
n | |
\sum | |
i=1 |
ic\in\Theta(nc+1)
n | |
\sum | |
i=1 |
1 | |
i |
\in\Theta(logen)
n | |
\sum | |
i=1 |
ci\in\Theta(cn)
n | |
\sum | |
i=1 |
log(i)c\in\Theta(n ⋅ log(n)c)
n | |
\sum | |
i=1 |
log(i)c ⋅ id\in\Theta(nd+1 ⋅ log(n)c)
n | |
\sum | |
i=1 |
log(i)c ⋅ id ⋅ bi\in\Theta(nd ⋅ log(n)c ⋅ bn)
\Sigma (2wx+w2)=x2
infty | |
\sum | |
i=1 |
-i2t | |
e |
\ldots