Summation Definition and 32 Discussions

In mathematics, summation is the addition of a sequence of any kind 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 [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, 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 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 1 + 2 + 3 + 4 + ⋯ + 99 + 100. Otherwise, summation is denoted by using Σ notation, where

{\textstyle \sum }
is an enlarged capital Greek letter sigma. For example, the sum of the first n natural integers can be denoted as




{\textstyle \sum _{i=1}^{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 expressions for the result. For example,







{\displaystyle \sum _{i=1}^{n}i={\frac {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.

View More On
  1. Sciencemaster

    I Finding ##\partial^\mu\phi## for a squeezed state in QFT

    I'm trying to apply an operator to a massless and minimally coupled squeezed state. I have defined my state as $$\phi=\sum_k\left(a_kf_k+a^\dagger_kf^*_k\right)$$, where the ak operators are ladder operators and fk is the mode function $$f_k=\frac{1}{\sqrt{2L^3\omega}}e^{ik_\mu x^\mu}$$...
  2. tworitdash

    A Analytical form of a summation

    I have a equation with a double sum. However, one of the variables in one of the sums comes from a stochastic distribution (Gaussian to be specific). How can I get a closed form equivalent of this expression? The U and Tare constants in the equation. $$ \sum_{k = 0}^{N_k-1} \bigg [ \big[...
  3. MevsEinstein

    B Is ##\sum^n_{k=0} 2k+1 = n^2## useful? Has it been found already?

    I was looking at the tiles of my home's kitchen when I realized that you can form squares by summing consecutive odd numbers. First, start with one tile, then add one tile to the right, bottom, and right hand corner (3), and so on. Can this be applied somewhere? And has someone found it already?
  4. A

    A Simplifying a double summation

    Is it possible to simplify the function below so that the sums disappear. $$\displaystyle g \left(x \right) \, = \, \sum _{j=-\infty}^{\infty} \left(-A +B \right) \sum _{k=-\infty}^{\infty} \frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(x -k \right)^{2}}{\sigma...
  5. C

    A Help required to sum an infinite series in a given equation

    Hi, I have a particular equation in a paper, wherein the author specifies an infinite series. The author has apparently found the sum of the series and calculated the equation. Can anyone please help me in understanding how to sum such a series. I have attached part of the paper with the...
  6. hilbert2

    A Summation formula from statistical mechanics

    I ran into this kind of expression for a sum that appears in the theory of 1-dimensional Ising spin chains ##\displaystyle\sum\limits_{m=0}^{N-1}\frac{2(N-1)!}{(N-m-1)!m!}e^{-J(2m-N+1)/kT} = \frac{2e^{2J/kT-J(1-N)/kT}\left(e^{-2J/kT}(1+e^{2J/kT})\right)^N}{1+e^{2J/kT}}## where the ##k## is the...
  7. L

    Summation for calc II

    Homework Statement (summation from 1 to 100) Σ (1/k) - (1/(k+1)) [/B] Homework Equations Σc = cn Σi = (n(n+1))/2[/B] The Attempt at a Solution I can only find summation equations for variables in the numerator. I'm not sure how to even start this problem. [/B]
  8. P

    B Quickest way to calculate a given summation

    How would you, personally, do this summation the quickest way?
  9. N

    I Problem when evaluating bounds....Is the result 1 or 0^0?

    Consider the summation ∑,i=0,n (t^(n-i))*e^(-st) evaluated from zero to infinity. You could break down the sum into: (t^(n))*e + (t^(n-1))*e + (t^(n-1))*e + ... + (t^(n-n))*e ; where e = e^(-st) To evaluate this, notice that all terms will go to zero when evaluated at infinity However, when...
  10. B

    Normalization of Wavefunction Integration

    Homework Statement [/B] Determine the value that A (assumed real) must have if the wavefunction is to be correctly normalised, i.e. the volume integral of |Ψ|2 over all space is equal to unity. Homework Equations Integration by parts (I think?) The Attempt at a Solution So, I've managed...
  11. D

    I Summation of 1^1+2^2+3^3+....+k^k

    Does that summatiom have a shorter representation at all? ##\sum_{n=1}^{k} n^n = ?## I guess it is not of the form of constant power series, but I could not find an alternative. Mentor note: made formula render properly
  12. G

    Proper usage of Einstein sum notation

    Homework Statement I'm dealing with some pretty complex derivatives of a kernel function; long story short, there's a lot of summations going on, so I'm trying to write it down using the Einstein notation, for shortness and hopefully reduction of errors (also for the sake of a paper in which I...
  13. JERRY-thechuha

    How to solve this partial derivative which includes a summation?

    I was reading a research paper, and I got stuck at this partial differentiation. Please check the image which I have uploaded. Now, I got stuck at Equation (13). How partial derivative was done, where does summation gone? Is it ok to do derivative wrt Pi where summation also includes Pi...
  14. J

    I Summation Equality

    I'm doing my first paper review and an equation is holding me up. I can't tell if I'm just missing something silly or if the author made a mistake. Given that: \sum_{n=1}^{N}s_{n} = 1 The author says that: \sum_{n=1}^{N}(s_{n} - \frac{1}{N})^{2} = \sum_{n=1}^{N}s_{n}^{2} - \frac{1}{N} I seem to...
  15. mooncrater

    I Upper Bound of a Summation

    There is this summation, that I've been trying to solve, but am not able to do so. It is : $$\sum\limits_{i=k}^{n} \frac {1}{(n-i)! m^{i-1}}$$ I would be happy to find it's upper bound too. So what I did was intensely naive. I made the denominator the minimum by making ##(n-i)! = 1## and...
  16. S

    Contravariant Four-gradient ESN in Wikipedia appears wrong

    Homework Statement I am self studying relativity. In Wikipedia under the four-gradient section, the contravariant four-vector looks wrong from an Einstein summation notation point of view. Homework Equations It states: E0∂0-E1∂1-E2∂2-E3∂3 = Eα∂α...
  17. S

    Finding the sum of a series by grouping

    Homework Statement Homework Equations Summation The Attempt at a Solution I know I could have simplified (3n-2)^3 +(3n-1)^3 -(3n)^3 and put the formulas in but I wonder is there any other method (I was thinking about grouping the terms, but to no avail) to work this out.
  18. Ben Wilson

    MATLAB Matlab syntax for 2-d fourier transform

    I have a function f(x,y) which i have defined in this way: a vector x and a vector y meshgrid[x,y] z= f(meshgrid[x,y]). how do i do a 2-d Fourier transform of f(x,y)? the transform must be done without using operations like fft, and must be done using summations written in the code.
  19. C

    I Summation algebra

    Hi, I'm working with series solutions of differential equations and I have come across something that has troubled me other courses as well. given that \begin{equation} \sum_{n=0}^{\infty} c_{n+2}x^n+e^{-x} \sum_{n=0}^{\infty}c_{n}x^n \\ \text{where}\\...
  20. F

    Summation Convention – Substitution Rule

    Hello. I'm new to this forum. I'm starting a PhD – it's going to be a big long journey through the jungle that is CFD. I would like to arm myself with some tools before entering. The machete is Cartesian Tensors. I know the rules regarding free suffix's and dummy suffixes, but I'm having...
  21. N

    Summation question

    I know n(n+1)/2 solves from 1... n by 1's Is there a formula where you can add up by 3's? Example: 3 + 6 + 9 + 12 ... n
  22. G

    Evaluate finite sum

    Homework Statement Find \sum\limits_{k=0}^{n}k^2{n\choose k}(\frac{1}{3})^k(\frac{2}{3})^{n-k} Homework Equations -Binomial theorem The Attempt at a Solution I am using the binomial coefficient identity {n\choose k}=\frac{n}{k}{{n-1}\choose {k-1}}: \sum\limits_{k=0}^{n}k^2{n\choose...
  23. nisler.1

    Biophysics Problem -- Summation Issues

    Homework Statement The average number of mRNAs in the cell at any time t is <m>(t) = Σ m * p(t). Sum over all the differential equations derived in a) in order to obtain a differential equation for <m>(t) Homework Equations So the differential equation I got in a) was dp/dt = (-kp * Pm) - (m *...
  24. C

    Error Propagation

    I tried posting this question in this forum a couple of weeks ago, but didn't get an answer to my question. I'm going to try posting it again using the formatting template so that it is hopefully clearer. I am also not sure if this is the right forum to be posting this in. It is a problem I ran...
  25. C

    Error Propagation - Reconciling Two Approaches

    Hi, I am trying to find the error propagated by calculating the sum of a set of mass flow rates collected over the same length of time. The sum of mass flow rates can be calculated with two approaches, since the collection time is the same for all of them. Approach (1) is adding up all of the...
  26. S

    What are generic terms for integration/summation parameters?

    This is not only a question strictly about mathematics, but in science or any other quantitative field in which there is an integration - or a summation that is like a discrete integration. [ A ] the parameter that is considered the input variable for the integration/summati - i.e., the x of dx...
  27. P

    Electrical Potential Energy of three quark system

    Homework Statement A proton is composed of three quarks: two "up" quarks, each having charge +2e/3, and one "down" quark, having charge -e/3. Suppose that the three quarks are equidistant from one another. Take the distance to be 3×10-15 m and calculate the potential energy of the subsystem of...
  28. Titan97

    Prove that [a/b]+[2a/b]+....+[(b-1)a/b]=(a-1)(b-1)/2

    Homework Statement Prove that $$\sum_{r=1}^{b-1}[\frac{ra}{b}]=\frac{(a-1)(b-1)}{2}$$ where [.] denotes greatest integer function and a & b have no common factors. Homework Equations ##n\le [n]<n+1## <x> denotes fractional part of x. 3. The Attempt at a Solution I first added and subtracted...
  29. N

    Calculus by Spivak, Chapter 2, Problem 6, Part 3

    In this problem, Spivak shows how to derive formulas to summations. They start by showing the method for 1^2 + 2^2 + ... + n^2 as follows: (k + 1)^3 - k^3 = 3k^2 + 3k + 1 Writing this formula for k = 1, 2, ..., n and adding, we obtain 2^3 - 1^3 = 3*1^2 + 3*1 + 1 3^3 - 2^3 = 3*2^2 + 3*2 + 1 ...
  30. A

    Adding increasing fractions without averaging numerators

    I'm interested in the following inequality (which may or may not be true) Theorem 1: ##( \sum_{i=1}^n \frac{a_i} {n}\ )( \sum_{i=1}^n \frac{1} {b_i}\ ) > \sum_{i=1}^n \frac{a_i} {b_i}\ ## Where ##n ≥ 2, a_1 < a_2 < ... < a_n## and ##b_1 < b_2 < ... < b_n##. My attempt at a proof: 1) When n =...
  31. JonnyMaddox

    Program that writes tensor equations out

    Hi, I'm looking for a program that spits out fully summed index equations. For example T_{ii} in, out comes T_{11}+T_{22}+... and so on, with Einstein summation convention.
  32. PWiz

    Product of two summations

    Homework Statement A and B are matrices and x is a position vector. Show that $$\sum_{v=1}^n A_{\mu v}(\sum_{\alpha = 1}^n B_{v\alpha}x_{\alpha})=\sum_{v=1}^n \sum_{\alpha = 1}^n (A_{\mu v} B_{v\alpha}x_{\alpha})$$ $$= \sum_{\alpha = 1}^n \sum_{v=1}^n(A_{\mu v} B_{v\alpha}x_{\alpha})$$ $$=...