Summation

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.{{cite book \sum_{i \mathop =m}^n a_i = a_m + a_{m+1} + a_{m+2} + \cdots + a_{n-1} + a_n where i is the "index of summation" or "dummy variable", ai is an indexed variable representing each term of the sum; m is the "lower bound of summation", and n is the "upper bound of summation". The "" under the summation symbol means that the index i starts out equal to m. The index, i, is incremented by one for each successive term, stopping when .{{efn|For a detailed exposition on summation notation, and arithmetic with sums, see {{cite book \sum_{i = 3}^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, 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 \sum a_i = \sum_{i = 1}^n 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, \sum_{0 \le k is an alternative notation for \sum_{k = 0}^{99} f(k), the sum of f(k) over all (integers) k in the specified range.{{cite book | editor-first = Kenneth H. | editor-last = Rosen | chapter-url = https://books.google.com/books?id=Xj4PEAAAQBAJ&pg=PA196 is the sum of \mu(d) over all positive integers d dividing n.

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 \sum_{i=\ell}^n \sum_{j=m}^k a_{i,j} . 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 \sum_{i=m}^n \sum_{j=m}^n a_{i,j} = \sum_{i,j=m}^n a_{i,j}.

The term **** is sometimes used when discussing the summation presented above. Contrast to the infinite series, the upper bound tends to infinity \sum_{i=m}^\infty a_i , 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 \sum_{i \ge 0} a_i .

Relatedly, the similar notation is used for the product of a sequence, 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, 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 n=m in the definition above, then there is only one term in the sum; if n=m-1, 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 \int to mark the sum of differentials (Latin: calculus summatorius), hence the S-shape.{{cite book |editor-first=Karl Immanuel |editor-last=Gerhardt |author-link=Joseph-Louis Lagrange |author-link=Joseph Fourier

Formal definition

Summation may be defined recursively as an operator, taking in a function and two natural numbers, as follows: :\sum : (\R^\R \times \N \times \N) \mapsto \R, such that: : :\sum_{i=a}^b g(i)=0, for b; : :\sum_{i=a}^b g(i)=g(b)+\sum_{i=a}^{b-1} g(i), for b \geqslant a.

Measure theory notation

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

:\sum_{k \mathop =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 \mu is the counting measure over the integers.

Calculus of finite differences

Given a function f that is defined over the integers in the interval [m, n], the following equation holds: :f(n)-f(m)= \sum_{i=m}^{n-1} (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: :f(n)-f(m)=\int_m^n f'(x),dx, where :f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h} is the derivative of f.

An example of application of the above equation is the following: :n^k=\sum_{i=0}^{n-1} \left((i+1)^k-i^k\right). Using binomial theorem, this may be rewritten as: :n^k=\sum_{i=0}^{n-1} \biggl(\sum_{j=0}^{k-1} \binom{k}{j} i^j\biggr).

The above formula is more commonly used for inverting of the difference operator \Delta, defined by: :\Delta(f)(n)=f(n+1)-f(n), where f is a function defined on the nonnegative integers. Thus, given such a function f, the problem is to compute the antidifference of f, a function F=\Delta^{-1}f such that \Delta F=f. That is, F(n+1)-F(n)=f(n). This function is defined up to the addition of a constant, and may be chosen as :F(n)=\sum_{i=0}^{n-1} 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)=n^k and, by linearity, for every polynomial function of n.

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.

and for any 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 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.

General identities

: \sum_{n=s}^t C\cdot f(n) = C\cdot \sum_{n=s}^t f(n) \quad (distributivity){{cite book : \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 (commutativity and associativity) : \sum_{n=s}^t f(n) = \sum_{n=s+p}^{t+p} f(n-p)\quad (index shift) : \sum_{n\in B} f(n) = \sum_{m\in A} f(\sigma(m)), \quad for a bijection σ from a finite set A onto a set B (index change); this generalizes the preceding formula. : \sum_{n=s}^t f(n) =\sum_{n=s}^j f(n) + \sum_{n=j+1}^t f(n)\quad (splitting a sum, using associativity) : \sum_{n=a}^{b}f(n)=\sum_{n=0}^{b}f(n)-\sum_{n=0}^{a-1}f(n)\quad (a variant of the preceding formula) : \sum_{n=s}^t f(n) = \sum_{n=0}^{t-s} f(t-n)\quad (the sum from the first term up to the last is equal to the sum from the last down to the first) : \sum_{n=0}^t f(n) = \sum_{n=0}^{t} f(t-n)\quad (a particular case of the formula above) : \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 (commutativity and associativity, again) : \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 (another application of commutativity and associativity) : \sum_{n=2s}^{2t+1} f(n) = \sum_{n=s}^t f(2n) + \sum_{n=s}^t f(2n+1)\quad (splitting a sum into its odd and even parts, for even indexes) : \sum_{n=2s+1}^{2t} f(n) = \sum_{n=s+1}^t f(2n) + \sum_{n=s+1}^t f(2n-1)\quad (splitting a sum into its odd and even parts, for odd indexes) : \sum_{n=s}^t \log_b f(n) = \log_b \prod_{n=s}^t f(n)\quad (the logarithm of a product is the sum of the logarithms of the factors) : C^{\sum\limits_{n=s}^t f(n) } = \prod_{n=s}^t C^{f(n)}\quad (the exponential of a sum is the product of the exponential of the summands) : \sum^{k}{m = 0}\sum^{m}{n = 0}f(m,n)=\sum^{k}{m = 0}\sum^{k}{n = m}f(n,m),\quadfor any function f from \mathbb{Z}\times\mathbb{Z}.

Powers and logarithm of arithmetic progressions

: \sum_{i=1}^n c = nc\quad for every c that does not depend on i : \sum_{i=0}^n i = \sum_{i=1}^n i = \frac{n(n+1)}{2}\qquad (Sum of the simplest arithmetic progression, consisting of the first n natural numbers.) : \sum_{i=1}^n (2i-1) = n^2\qquad (Sum of first odd natural numbers) : \sum_{i=0}^{n} 2i = n(n+1)\qquad (Sum of first even natural numbers) : \sum_{i=1}^{n} \log i = \log (n!)\qquad (A sum of logarithms is the logarithm of the product) : \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 (Sum of the first squares, see square pyramidal number.) : \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 (Nicomachus's theorem)

More generally, one has Faulhaber's formula for p1 : \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}, where B_k denotes a Bernoulli number, and \binom p k is a binomial coefficient.

Summation index in exponents

In the following summations, a is assumed to be different from 1.

: \sum_{i=0}^{n-1} a^i = \frac{1-a^n}{1-a} (sum of a geometric progression) : \sum_{i=0}^{n-1} \frac{1}{2^i} = 2-\frac{1}{2^{n-1}} (special case for ) : \sum_{i=0}^{n-1} i a^i =\frac{a-na^n+(n-1)a^{n+1}}{(1-a)^2} (a times the derivative with respect to a of the geometric progression) : \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} :::(sum of an arithmetico–geometric sequence)

Binomial coefficients and factorials

Main article: 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

: \sum_{i=0}^n {n \choose i}a^{n-i} b^i=(a + b)^n, the binomial theorem : \sum_{i=0}^n {n \choose i} = 2^n, the special case where : \sum_{i=0}^n {n \choose i}p^i (1-p)^{n-i}=1, the special case where , which, for 0 \le p \le 1, expresses the sum of the binomial distribution : \sum_{i=0}^{n} i{n \choose i} = n(2^{n-1}), the value at of the derivative with respect to a of the binomial theorem : \sum_{i=0}^n \frac{n \choose i}{i+1} = \frac{2^{n+1}-1}{n+1}, the value at of the antiderivative with respect to a of the binomial theorem

Involving permutation numbers

In the following summations, {}{n}P{k} is the number of k-permutations of n. : \sum_{i=0}^{n} {}{i}P{k}{n \choose i} = {}{n}P{k}(2^{n-k}) : \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)} : \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}^+, where and \lfloor x\rfloor denotes the floor function.

Others

: \sum_{k=0}^{m} \binom{n+k}{n} = \binom{n+m+1}{n+1} : \sum_{i=k}^{n} {i \choose k} = {n+1 \choose k+1} : \sum_{i=0}^n i\cdot i! = (n+1)! - 1 : \sum_{i=0}^n {m+i-1 \choose i} = {m+n \choose n} :\sum_{i=0}^n {n \choose i}^2 = {2n \choose n} :\sum_{i=0}^n \frac{1}{i!} = \frac{\lfloor n!; e \rfloor}{n!}

Harmonic numbers

: \sum_{i=1}^n \frac{1}{i} = H_n\quad (the nth harmonic number) : \sum_{i=1}^n \frac{1}{i^k} = H^{(k)}_n\quad (a generalized harmonic number)

Growth rates

The following are useful approximations (using theta notation):

: \sum_{i=1}^n i^c \in \Theta(n^{c+1}) for real c greater than −1 : : \sum_{i=1}^n \frac{1}{i} \in \Theta(\log_e n) (See Harmonic number) : : \sum_{i=1}^n c^i \in \Theta(c^n) for real c greater than 1 : : \sum_{i=1}^n \log(i)^c \in \Theta(n \cdot \log(n)^{c}) for non-negative real c : : \sum_{i=1}^n \log(i)^c \cdot i^d \in \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 \in \Theta (n^d \cdot \log(n)^c \cdot b^n) for non-negative real b 1, c, d

Notes

References

Bibliography

• {{cite book |author-link=Florian Cajori

References

1. "Summation Notation".
2. ''Handbook of Discrete and Combinatorial Mathematics'', Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, {{isbn. 0-8493-0149-1.