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.
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?
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 $$f_k=\frac{1}{\sqrt{2L^3\omega}}e^{ik_\mu x^\mu}$$...
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...
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...
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
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
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.
$$ \sum_{k = 0}^{N_k-1} \bigg [ \big[...
I was looking at the tiles of my home's kitchen when I realized that you can form squares by summing consecutive odd numbers. First, start with one tile, then add one tile to the right, bottom, and right hand corner (3), and so on. Can this be applied somewhere? And has someone found it already?
1. I have come across a few times I would like a more straightforward way to run a summation function on a summation function. I don't have the educational groundwork to know if there is another way to do this or a good technique to simplify these problems.
example...
Greetings!
I want to caluculate the summation of this following serie
I started by removing the 4 by
and then
and I thought of the taylor expansion of
Log(1-x)=-∑xn/n but as the 2 is not inside (-1,1) I couldn´t use it
any hint?
thank you!
Best !
Is it possible to simplify the function below so that the sums disappear.
$$\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...
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...
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?
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...
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.
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?
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...
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...
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
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##...
Let $x_1,\,x_2,\,\cdots,\,x_{2014}$ be the roots of the equation $x^{2014}+x^{2013}+\cdots+x+1=0$. Evaluate $\displaystyle \sum_{k=1}^{2014} \dfrac{1}{1-x_k}$.
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...
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...
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...
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?
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...
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?
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...
Hello,
I want to convert a summation in reciprocal space and I am unsure about the integration volume. I have started with the formula:
$$\sum_{\vec{k}} \rightarrow \frac{V_{k}}{(2\pi)^{3}}\int\int\int \mathrm{d}V_{k}$$
where:
$$\mathrm{d}V_{k} = k^{2}\mathrm{d}k...
I can’t understand the term Summation of (j tj). Are we multiplying j and tj. But the text is not talking about multiplying, it says”Total processing time
Somebody please guide me.
Zulfi.
Attempted Solution:
$$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})$$
Beyond this, though, I am quite lost. I know I am very close to the answer, but seeing this many terms can become fairly...
I have an identity
$$\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...
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...
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...
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?
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...
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...
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...
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...
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)
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.
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?
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+...
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]