Summation Definition and 610 Threads

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

how to write a summation with decreasing index say for adding from index 1 to n for $x_k$ we write $\sum^{n}_{k=1}x_k$. how do we write the above for index to go from n to 1 down wards

In Chapter 7: Hamilton's Principle, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, page 258-259, we have the following equations: 1. Upon squaring Equation (7.117), why did the authors in the first term of Equation (7.118) are summing over two...

I have recently been investigating summing divergent series and zeta function regularization's relation to dimensional re-normalization. Making some progress, but it is a bit slow despite literature being available...

Homework Statement "Prove that ##\sum_{n=0}^\infty s^n e^{-\lambda} \frac{\lambda^n}{n!}\sum_{m=0}^\infty s^m e^{-\mu}\frac{\mu^m}{m!}=\sum_{m+n=0}^\infty s^{n+m} e^{-(\lambda+\mu)} \frac{(\lambda + \mu)^{m+n}}{(m+n)}!## Homework Equations Binomial theorem: ##(x+y)^n=\sum_{k=0}^n x^ky^{n-k}##...

Homework Statement Prove by induction that ##\sum\limits_{k=1}^{2n} \frac{1}{k(k+1)} = \frac{2n}{2n+1}## 2. The attempt at a solution First I showed that it is true for ##n=1## and ##n=2##. Then, assuming it is true for all ##n##, I attempt to show that it is true for ##n+1##...

Is (Tii)2 equivalent to (∑i = 1nTii)2? That is, when you encounter parentheses with Einstein summation, you perform the summation first and then apply any mathematical operations indicated by the parentheses? The solutions manual gives a solution to a problem I've been working out seems to...

Hi All Been investigating lately ways to sum ordinarily divergent series. Looked into Cesaro and Abel summation, but since if a series is Abel Mable it is also Cesaro sumable, but no, conversely,haven't worried about Cesaro Summation. Noticed Abel summation is really a regularization...

Homework Statement I know that by definition Γijkei=∂ej/∂xk implies that Γmjk=em ⋅ ∂ej/∂xk (e are basis vectors, xk is component of basis vector). Can I write it in the following form? Γjjk=ej ⋅ ∂ej/∂xk Why or why not? Homework EquationsThe Attempt at a Solution

Homework Statement Assume that you want to the derivative of a vector V with respect to a component Zk, the derivative is then ∂ViZi/∂Zk=Zi∂Vi/∂Zk+Vi∂Zi/∂Zk = Zi∂Vi/∂Zk+ViΓmikZm Now why is it that I can change m to i and i to j in ViΓmikZm?

n ∑ 3 k=0 How does this make sense when k=0?

In definite Integration simply we find out the area under the curve.If I am not wrong the processing is like that we divide the area in infinite numbers of rectangles and then find out the summation of all rectangles area. My Question is here, how it is possible to add infinity? assume you have...

Hello, I have the following product, and I am looking for a summation equivalent \prod_{k=1}^K\left(1-\frac{1}{x_k+1}\right) Is this doable? I tried to use partial fraction but got nowhere! Thanks in advance

Homework Statement I am unsure as to how to write the dot product in terms of the summation notation? May you please explain? Homework EquationsThe Attempt at a Solution

Hi, I have this random variable ##\beta=\sum_{k=1}^K\alpha_k##, where ##\{\alpha_k\}_{k=1}^{K}## are i.i.d. random variables with CDF ##F_{\alpha}(\alpha)=1-\frac{1}{\alpha+1}## and PDF ##\frac{1}{(1+\alpha)^2}##. I want to find the CDF of the random variable ##\beta##. So, I used the Moment...

Homework Statement Show that the sequence of partial sums s_{n} = 1+\sum_{i=1}^{n} \left(\prod_{k=1}^{i}\left( \frac{1}{2} + \frac{1}{k}\right)\right) converges, with n\in \mathbb{N}\cup \{0\} Homework EquationsThe Attempt at a Solution [/B] So we want to find \lim_{n\to\infty} s_{n} =...

Hi, I am trying to simplify a double summation and was wondering if anyone would be able to help me. The sum is $$ \sum_{i=1}^{n-1} \sum_{j=i+1}^n a_i a_j $$ Is it possible to simplify it down and maybe lose one of the sigmas? Thank you in advance :)

Homework Statement Through out my linear algebra book, this weird summation sign has started appearing, and I haven't been able to find anything on it online. Can someone please explain how I'm suppose to read this: Homework Equations The Attempt at a Solution Now in the first case, I could...

Homework Statement Show that ##\sum_{k=2}^\infty d_k## converges to ##\lim_{n\to\infty} s_{nn}##. Homework Equations I've included some relevant information below: The Attempt at a Solution So far I've managed to show that ##\sum_{k=2}^\infty |d_k|## converges, but I don't know how to move...

I am asking on the spur, so there has not been too much thought put into it, but how would we classify a series summation such as $$ \sum_{i=0}^{n} 2^{2^i} ~ ?$$ It does not feel to be geometric, nor that it can be made to be geometric. In general, the function xx does not look like it bears a...

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...

Homework Statement In the calculation in high temperatures of ##Z_{rot} = (\sum_{j=0}^\infty (2j+1)\exp{j(j+1)\theta_{rot}/T})^N##; they use Euler summation formula: $$\sum_{n=0}^\infty f(n) = \int_0^\infty f(x)dx+\frac{1}{2}f(0)-\frac{1}{12}f'(0)+\frac{1}{720}f^{(3)}(0)+\ldots$$ for ##f(x) =...

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...

Homework Statement The question asks us to write down the matrix represented by the following summation. 2. Homework Equations The question summation... $$\sum_{j,k=1}^{3} a_{ij}b_{jk}x_{k}$$ The Attempt at a Solution $$ \sum_{j,k=1}^{3} a_{ij}b_{jk}x_{k} =...

What does the equation ζ(−1) = −1/12 represent precisely? It's impossible for that to be the sum of all natural numbers. And it is also mentioned in all the maths articles that the 'equal to' in the equation should not be understood in a traditional way. If so, then why wikipedia article...

I am trying to manipulate this summation such that I have a summation of a function of r only by itself somewhere: \sum_{r=1}^{\infty}e^{-B⋅r} This could be rewritten: \sum_{r=1}^{\infty}\left(e^{-r}\right)^B or \sum_{r=1}^{\infty}\left(e^{-B}\right)^r What I would like is: f(r)g(B) or...

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[/size]

I have recently been doing some work that involves long, messy manipulations with lots of geometric series. A typical such series, which would only be one of a number of such terms in a formula, is: $$ \sum_{t=h+1}^{T-h} \left(1-(1-\theta)^{T-t-h+1}\right) \\ $$ It's not difficult to simplify...

Homework Statement Homework Equations Summs: $$1+2+3+...+n=\frac{n(n+1)}{2}$$ $$1^2+2^2+3^2+...+n^2=\frac{n(n+1)(2n+1)}{6}$$ The Attempt at a Solution $$\Delta x=\frac{b}{n}$$ $$S_n=f\left( \frac{\Delta x}{2} \right)\Delta x+f\left( \Delta x+\frac{\Delta x}{2} \right)\Delta x+...+f\left(...

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...

Homework Statement Why, in: $$\frac{\sqrt{1}+\sqrt{2}+...+\sqrt{n}}{n^{3/2}}$$ There is ##~n^{3/2}## in the denominator? Homework Equations The Attempt at a Solution it should be: $$S_n=\sqrt{c_1}\Delta x+\sqrt{c_2}\Delta x+...=\Delta x\cdot \sqrt{\Delta x}+\Delta x\cdot \sqrt{2\Delta...

Homework Statement "Given: ##sin(t)=Σ\frac{(-1)^nt^{2n+1}}{(2n+1)!}## Prove: ##L[sin(t)]=\frac{1}{s^2+1}##." Homework Equations ##∑ar^k=\frac{a}{1-r}## The Attempt at a Solution...

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...

When taking the superposition of wavefunctions with definite values of any observable (I'll just use momentum, but I am assuming it would work for any variable), I have seen the integral be used: ##\psi = \int_{-\infty}^{\infty}\phi(k)e^{ikx}dk## and the sum be used: ##\psi =...

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...

Homework Statement Picture has been uploaded with question. Homework EquationsThe Attempt at a Solution I have literally never seen anything like this in my life. I'm in mathematical physics. I looked up de Moivre's formula and I guess this comes from a course in complex variables? I don't...

Homework Statement I'm give the following summation of functions and I have to see where it converge. $$\sum_{n = 1}^{\infty} \frac{(3 arcsin x)^n}{\pi^{n + 1}(\sqrt(n^2 + 1) + n^2 + 5)}$$ Homework EquationsThe Attempt at a Solution Putting ##3 arcsin x = y##, I already see that with the...

Dear Friends So, i have this special case where i have to do a differentiation and summation. Please check the following. Is it okay ?? Or, i how should i proceed with this ?

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...

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. https://en.wikipedia.org/wiki/Four-vector Homework Equations It states: E0∂0-E1∂1-E2∂2-E3∂3 = Eα∂α...

Homework Statement Find $$\sum_{k=1}^{64} {64 \choose k} 64k$$ Homework Equations Not sure The Attempt at a Solution Please give me hint how to start doing this question

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.

Hi PF! As outlined in my book ##\delta_{ij} \delta_{jk} = \delta_{ik}## but don't we sum over repeated indices (and the ##j## is repeated)? Can someone explain why we do not sum in this situation? Thanks!

Homework Statement Using change of summation index show that: $$\sum_{k=1}^{n} (k + 1)^3 - \sum_{k=1}^{n} (k-1)^3 = (n + 1)^3 + n^3 - 1$$Hence show that: $$\sum_{k=0}^{n} k^2 = \frac{n}{6} (n + 1)(2n + 1)$$ 2. The attempt at a solution For the first part I changed the summation index like...

Is there a way of writing summation(s) to obtain the extended binomial coefficients? i.e., Considering the expansion of (1+x+x^2+x^3+...+x^N)^M can we write expressions (presumably involving summation and/or product notation) for the coefficients (on x^j in the expansion of the above, for each...

I was recently researching into some string theory when i came across the following summation: The sum of all natural numbers is -1/12, now I'm still wrapping my head around the context of the application within critical string dimensions, but is this summation valid? And if not, why it being...

Homework Statement True or False? If A is an n × n matrix, P is an n × n invertible matrix, and B = P −1AP, then a11 + a22 + . . . + ann = b11 + b22 + . . . + bn Homework Equations Diagnolization, similar matrixes The Attempt at a Solution the question is asking if the summation of the main...

Is ∑f from a to b the same as ∑f from b to a? In other words, does the order of summation matter?