# What is Summation: Definition and 626 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

i
=
1

n

i
.

{\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,

i
=
1

n

i
=

n
(
n
+
1
)

2

.

{\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.

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1. ### I Question on an infinite summation series

Is the infinite series ##\sum_{n=1,3,5,...}^\infty \frac {1} {n^6}## somewhat related to the Riemann zeta function?The attached image suggest the value to be inverse of the co-efficient of the series.Is there any integral representation of the series from where the series can be evaluated?
2. ### 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 $$f_k=\frac{1}{\sqrt{2L^3\omega}}e^{ik_\mu x^\mu}$$...
3. ### B Beginner Einstein Notation Question On Summation In Regards To Index

So, I have recently been trying to learn how to work with tensors. In doing this, I have come across Einstein Notation. Below is my question. $$(a_i x_i)_{e}= (\sum_{i=1}^3 a_i x_i)_r=(a_1 x_1+a_2 x_2+a_3 x_3)_r$$; note that the following expression is in three dimensions, and I use the...
4. ### I From Einstein Summation to Matrix Notation: Why?

I know that if ##\eta_{\alpha'\beta'}=\Lambda^\mu_{\alpha'} \Lambda^\nu_{\beta'} \eta_{\alpha\beta}## then the matrix equation is $$(\eta) = (\Lambda)^T\eta\Lambda$$ I have painstakingly verified that this is indeed true, but I am not sure why, and what the rules are (e.g. the ##(\eta)## is in...
5. ### How to prove this using Abel's summation formula?

Before I apply/use the Abel's summation formula, how should I find ## f(x) ##?
6. ### Use Euler's summation formula to prove the following....

Proof: Let ## x\geq 2 ##. Then ## \frac{d}{dt}(\frac{ \log {t}}{t^3})=\frac{1-3\log {t}}{t^4} ##. By Euler's summation formula, we have that ## \sum_{n\leq x}\frac{ \log {n}}{n^3}=\int_{1}^{x} \frac{\log {t}}{t^3}dt+\int_{1}^{x} (t-[t])(\frac{1-3\log {t}}{t^4})dt+(x-[x])\frac{log {x}}{x^3} ##...
7. ### I Einstein summation convention in QM

Hi For an operator A we have Aψn = anψn ; the matrix elements of the operator A are given by Amn= anδmn My question is : is this an abuse of Einstein summation convention or is that convention not used in QM ? Thanks
8. ### Maple Summation of a Finite Series: Seeking the Sum with MAPLE or Other Software

Dear Colleagues I hope this post belongs here in calculus. It concerns a finite series for which I am seeking the sum. I tried using MATHEMATICA which didn't accept it. Perhaps if someone has Maple or any other software who can do it. Here it is attached. I shall be most grateful

14. ### A Klein Gordon Lagrangian -- Summation question

Klein Gordon Lagrangian is given by \mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^2\phi^2 I saw also this link https://www.pas.rochester.edu/assets/pdf/undergraduate/the_free_klein_gordon_field_theory.pdf Can someone explain me, what is...
15. ### Trouble with a double summation

the inner sum is just adding 1/365 n-i number of times. so ##\frac{(n-i)}{365}## the outer sum adds over the index i, so I thought the expression is equal to ##\frac{(n-1)n-(n-1)!}{365}## but it's obviously not equal to that. where did I go wrong?
16. ### I Einstein summation convention confusion

Hi If i have a vector r = ( x1 , x2 , x3) then i can write r2 as xixi where the i is summed over because it occurs twice. Now is xixi the same as xi2 ? I have come across an example where they are used as equivalent but i am confused because xi2 seems to be the square of just one component of r...
17. ### A Non-Linear Theory: Summation Meaningful in Einstein Gravitation?

In the famous book, Gravitation, by Misner, Thorne and Wheeler, it talks about the stress-energy tensor of a swarm of particles (p.138). The total stress-energy is summed up from all categories of particles. Is summation meaningful in the non-linear theory of Einstein gravitation? Thanks.
18. ### I Summation notation and general relativity derivatives

Does $$\partial^\beta(g_{\alpha\beta}A_\mu A^\mu)$$ mean the same as $$\frac {\partial (g_{\alpha\beta}A_\mu A^\mu)}{\partial A^\beta} ?$$ If not could someone explain the differences?
19. ### A Summation simpilification process

Hello, Here is my summation: sum(1/(25+n*B)),n=0 to (N/2)-1:=A where A is between .01 to 2, N is between 10 to 2000 and I need to find the B for different values of N. I calculate this summation online (check here) But the Digamma function makes the output function complex and it is not easy to...
20. ### MHB How Can We Prove the Summation Inequality for a Given Sequence?

Assume that $x_1,\,x_2,\,\cdots,\,x_n \ge -1$ and $\displaystyle \sum_{i=1}^n x_i^3=0$. Prove that $\displaystyle \sum_{i=1}^n x_i \le \dfrac{n}{3}$.
21. ### A How to optimize a parameter that is the index of a summation?

Hi everyone! So, the problem I'm having has more to do with "how to pose the problem to solve it in some software as Matlab or similar". I have experimentally measured values ##\varepsilon_{exp}^i## with ##i=1,\cdots,6##, that is, I have 6 detectors. Then, I know (from a Monte Carlo...
22. ### Integral as approximation to summation

Writing down several terms of the summation and then doing some simplifying, I get: $$\sum_{r=1}^n \frac{1}{n} \left(1+\frac{r}{n} \right)^{-1}= \frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...\frac{1}{2n}$$ How to change this into integral form? Thanks
23. ### I What does it mean to summate from 1 upwards to something less than 1?

Hello, I was finding the average value of the expression ##(1-1/n^2)## for values from 1 to infinity by evaluating the limit as N→∞ for: ## \displaystyle\sum_{n=1}^{N} (1-1/n^{2})/N ## and got what I expected, ##1## What I didn't expect was to find that the general solution ##1-H_N^{(2)}/N##...
24. ### MHB Summation Challenge: Evaluate $\sum_{k=1}^{2014}\frac{1}{1-x_k}$

Let $x_1,\,x_2,\,\cdots,\,x_{2014}$ be the roots of the equation $x^{2014}+x^{2013}+\cdots+x+1=0$. Evaluate $\displaystyle \sum_{k=1}^{2014} \dfrac{1}{1-x_k}$.
25. ### A Ramanujan Summation, Variations of

Hi, I've seen several videos and documents that state that "the sum of all natural numbers is equal to -1/12". The "proof" in general is using ramanjuan summation and analytic continuation of the riemann function. In this proof, the election of the riemann function in order to perform the...
26. ### MHB Summation and product notation rules

As per the image, I am supposed to select all the valid statements. Apparently I'm only partially correct, and so I took another look at the statements. I believe the third statement is wrong, since c * (a_m*a_{m+1}*a_{m+2}*...*a_n) =/= (c*a_m)(c*a_{m+1})(c*a_{m+2})*...*(c*a_n) Thus there...
27. ### MHB How Do You Solve a Geometric Sum with Alternating Signs?

Hey! I'm stuck again and not sure how to solve this question been at it for a few hours. Any help is appreciated as always. Q: (1) Let the sum S = 3- 3/2 + 3/4 - 3/8 + 3/16 - 3/32 +...- 3/128. Determine integers a , n and a rational number k so that...(Image) (2 )And then calculate S using...
28. ### MHB Double summation: inner index = function of outer index

Here N, a, and b are integer constants. M is also an integer but changes for every value of x, which makes the index of the second summation dependent on the first. The problem is the relationship M(x) is analytically difficult to define. Is there a way to solve/simplify this expression?
29. ### Finding a formula for the following summation

ive used google https://www.google.com/search?q=n%5E2+%2B+(n-2)%5E2+%2B+..&rlz=1C1SQJL_enUS890US890&oq=n&aqs=chrome.1.69i59l3j69i57j0j69i61j69i60l2.4719j0j7&sourceid=chrome&ie=UTF-8 and I was surprised that there was no relevant formula found How do I get/even begin to get the formula for...
30. ### B A summation that "feeds back into itself"

Hi everyone! I need to use sigma notation to build a summation that "feeds back into itself". By that I mean that it should model a sum whose terms are f(x) + f(f(x)) + f(f(f(x))) and so on. How would I do this?
31. ### MHB Summation Challenge: Evaluate $\sum_{n=0}^\infty \frac{16n^2+20n+7}{(4n+2)!}$

Evaluate $\displaystyle \sum_{n=0}^\infty \dfrac{16n^2+20n+7}{(4n+2)!}$.
32. ### Does Summation Over n from -∞ to +∞ in Quantum Mechanics Equal Ψ(x)?

I have a (trivial) question regarding summation notation in Quantum mechanics. Does ∑cnexp(iknx) = Ψ(x) imply that n ranges from -∞ to +∞ (i.e. all possible combinations of n)? i.e. n ∞ ∑exp(iknx) -∞ I believe it does to be consistent with the Fourier series in terms of complex exponentials...

36. ### How Can I Simplify and Solve the Einstein Summation Convention Problem?

Attempted Solution: $$a_i \, b_j \, c_k \, \epsilon_{ij \ell} \, \epsilon_{3k \ell}$$ $$a_i\, b_j\, c_k\, (\delta_{i3} \, \delta_{jk} - \, \delta_{ik}\, \delta_{j3})$$ Beyond this, though, I am quite lost. I know I am very close to the answer, but seeing this many terms can become fairly...
37. ### Deriving an identity using Einstein's summation notation

I have an identity $$\vec{\nabla} \times (\frac{\vec{m} \times \hat{r}}{r^2})$$ which should give us $$3(\vec{m} \cdot \hat{r}) \hat{r} - \vec{m}$$ But I have to derive it using the Einstein summation notation. How can I approach this problem to simplify things ? Should I do something like...
38. ### I Order of summation in series with multiple indices

Can someone help me understand why what I wrote is correct? That is: If I have a sequence with double indices and if the summation of the elements modules of this sequence converges (less than infinite) than it does not matter how I make this sum (second line) they are going to be always the...
39. ### I Summation Convention in Einstein Notation

I got another basic question: should the summation in einstein notation start from first occurance of index or in beginning of equation? For eampledoes this equation ##R_{\alpha \beta }={R^{\rho }}_{\alpha \rho \beta }=\partial _{\rho }{\Gamma ^{\rho }}_{\beta\alpha }-\partial _{\beta }{\Gamma...
40. ### B Einstein summation notation, ambiguity?

If I see ##f(x_ie_i)## I assume it means ##f(\Sigma x_ie_i)## (summing in the domain of f) but what if I instead wanted to write ##\Sigma f(x_ie_i)## (summing in the range)? Is there a way to distinguish between these in Einstein’s summation notation?
41. ### 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...

43. ### How to solve summation equations with different indices

To analyze the LHS of this equation, I used (k-1) , k and (K+1) to get ## \frac {(-1)^{k-1} } { (k-1)} \ . \frac {(-1)^k} { (k)} \ . \frac {(-1)^{k+1} } { (k+1)} \ ## Nothing cancels out in these terms and the sign of each term is the opposite of the previous term. I calculated...
44. ### Evaluating Summation of an Infinite Series

Evaluate ##\lim_{n \rightarrow +\infty} \frac {1} {n} [(\frac {1}{n})^{1.5} + (\frac {2}{n})^{1.5} +(\frac {3}{n})^{1.5}+ (\frac {4}{n})^{1.5}+...+(\frac {n}{n})^{1.5}]## Hello. So I'm solving this question at the moment. I know I'm supposed to find out the summation of this before being able...
45. ### 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...
46. ### Calculating Integrals Using the Fundamental Theorem of Calculus

Here, width of first bar, y=x^2=a^2 y=x^2=(a+Δx)^2 height of nth bar=y=(a+(N-1)Δx)^2 Total area,I={a^2+(a+Δx)^2+(a+2Δx^2)+...+[a+(N-1)Δx]^2}Δx I={Na^2 + 2aΔx +...} I can't seem to get forward to get the required result which is 1/3(b^3-a^3)
47. ### Is There a Formula for These Summation Problems?

I look though some algebra and calculus books but I didn't see any formula for this some, and I am stuck here. I can just represent it in a notation but I cannot think a formulation to obtain the result. ##\sum_{k=1}^{n=5}=\frac {n!}{n!k} ## Thank you.
48. ### I Changing Summation to Integral

This is the text from Reif Statistical mechanics. In the screenshot he changes the summation to integral(Eq. 1.5.17) by saying that they are approximately continuous values. However,I don't see how. Can anyone justify this change?
49. ### B Sum of Series 1/n: Is it Infinity?

I've seen the proof that the sum of 1/n for = 1 to infinity is infinity (which still blows my mind a little). Is the sum of 1/nn for n = 1 to infinity also infinity? i.e, 1 + 2/4 + 3/27 + 4/256+...
50. ### Efficient Methods for Solving Summation Equations: Σ(1/k) - (1/(k+1))

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]