Summation Definition and 610 Threads

Homework Statement Let f(z) = \sum_{n =-\infty}^{\infty} e^{2 \pi i n z} e^{- \pi n^2}. Show that f(z+i) = e^{\pi} e^{-2\pi i z}f(z). Homework Equations Nothing specific I can think of; general complex analysis/summation techniques. The Attempt at a Solution f(z+i) = \sum_{n...

Problem stated: Let \(a_1, a_2, ... , a_n\) be \(n\) positive numbers. Find the maximum of $$\sum_{i=1}^{n}a_ix_i$$ subject to the constraint $$\sum_{i=1}^{n}x_i^2=1$$. I honestly have not much of an idea of how to go about solving this. If I use lagrange multipliers which I think I am supposed...

prove the identity $$\nabla\times(f\cdot\vec{v})=(\nabla f) \times \vec{v} + f \cdot \nabla \times \vec{v}$$ I can do the proof with normal vector calculus, but I am in a tensor intensive course and would like to do this with einstein summation notation, but am having some trouble since I am...

Homework Statement Prove or disprove that: \frac{{\sum_{i=0}^{ord_N (2) - 1}} (2^i \bmod N)}{N} Is equal to the number of odd residue classes of 2^x \bmod N for all odd numbers N greater than 1. Homework Equations Residue Classes are the residues that are generated by a function...

Dear members, see attached pdf file.Can you help me to prove this formulas. Thank you Belgium 12 This is not homework.I'm 68 and retired.

i need to find a series of numbers up to n that will add up to kn x1 + x2 + x3+ ... + n = kn where k is a constant. this is part of a long complex problem once this sum is found it will finally be solved.

Homework Statement (\sin x) (\cos x)^{n - 1} + (\sin x) (\cos x)^{\frac{n - 1}{2}} + (\sin x) (\cos x)^{\frac{\frac{n - 1}{2}}{2}} + (\sin x) (\cos x)^{\frac{\frac{\frac{n - 1}{2}}{2}}{2}} ... n is an odd number and the series ends when (n-1)/2^k = 1, and the last term ends up being sin(x)...

When Planck first derived the concept of quantization, he treated the integrand for average energy =$\int_{0}^{\infty} \epsilon*P(\epsilon) d\mu$ , where $P(\epsilon)$ is the Boltzmann distribution as a summation nh\mu, and derived the Planck law. While when we use it to derived the...

Homework Statement Let v1, v2, v3 be a sequence and let un=nvn-(n+1)vn+1 for n= 1,2,3... find \sumun from n=1 to N. Homework Equations The Attempt at a Solution Began with method of differences and arrived at Sn= v1-(n+1)vn+1

[FONT=times new roman]Please consider the following equation: $\displaystyle \sum_{k=1}^{n}\cos^4\left(\frac{k\pi}{2n+1} \right)=\frac{6n-5}{16}$ [FONT=times new roman] For this particular equation, which I am trying to prove is true, I have found no way to crack it, even if I let $n=2$ and...

I think I get summation notation when when there are more numbers than variables 6 Ʃ i/6 <---I can figure that out. i=1 But I'm confused on how to find what this equals: N Ʃ (i^3)/(N^3) = ? i=1 How do you add something N times? ...I could deal with a number like 6, but I'm...

It has been a while since I've had to figure out summation notation. Would you please look through my solutions, and tell me if they're correct? Thank you so much! :) 1a. 6 Ʃ 1/6 = ? i=1 1/6 + 1/6 + 1/6 + 1/6 + 1/6 +1/6 = 6/6 = 1 What makes me doubt my answer is that it seems like...

Homework Statement Write out c_{j}x_{j}+c_{k}y_{k} in full, for n=4. Homework Equations The Attempt at a Solution So I figure we have to sum over both j and k. So the answer I obtained is: (c_1x_1+c_1y_1)+(c_1x_1+c_2y_2)+(c_1x_1+c_3y_3)+(c_1x_1+c_4y_4)+...

Homework Statement Find the sum of the series \displaystyle S_1=1 + \frac{x^3}{3!}+\frac{x^6}{6!}+\,\dots Can't seem to get the bit above to show up nicely, should be 1+x^3/3! +x^6/6! +... Sorry! Homework Equations In a prior part of the question I had to find the complex roots of z3-1=0...

The summation convention for Tensor Notation says, that we can omit the summation signs and simply understand a summation over any index that appears twice. So consider a 3X3 matrix A whose elements are denoted by aij, where i and j are indices running from 1 to 3. Now consider the...

Can anyone tell me what is the problem with this Mathematica code? Nmax = 10; Mmax = 10; A = 4/Pi^2*Integrate[x*Sin[n*x]*Sin[m*y], {x, 0, Pi}, {y, 0, Pi}]; B = 4/Pi^2*Integrate[Sin[n*x]*Sin[m*y], {x, 0, Pi}, {y, 0, Pi}]; u[x_, y_, t_] = Sum[Sin[n*x]* Sin[m*y] (A*Cos[(n^2 + m^2)*t] +...

Homework Statement What is easiest way to summate \sum^{\infty}_{n=1}J_n(x)[i^n+(-1)^ni^{-n}] where ##i## is imaginary unit. Homework Equations The Attempt at a Solution I don't need to write explicit Bessel function so in sum could stay C_1J_(x)+C_2J_2(x)+... Well I see that...

On another site, a user asked for help showing: $\displaystyle \sum_{k=0}^{2499} \frac{1}{\sqrt{4k+1}+\sqrt{4k+3}}>24$ The first respondent asked if the OP was familiar with mathematical induction. The reply was that induction was the topic of the next chapter in her course. Another suggested...

1. Homework Statement ∑ i=1 to n1+(1/i2)+(1/(1+i)2)−−−−−−−−−−−−−−−−−−−−√ = n(n+2)/n+1 2. The attempt at a solution First I did the base case of p(1) showing 3/2 on the LHS equals the 3/2 on the RHS. Then I assumed p(k) and wrote out the formula with k in it. Then prove p(k+1)= p(k)+...

I need to simplify this expression and I don't know how to deal with the factorials in the sum e^{-(\lambda + \mu)}\sum_{k=0}^w \frac{\lambda^k \mu^{(w-k)}}{k!(w-k)!} Can anybody give me a hint on how to sum over the factorials?

11.2 nmh{2000} Find a formula for the sum of $n$ terms Use the formula to find the limit as $n\to\infty$ $\displaystyle \lim_{n\to\infty} \sum\limits_{i = 1}^{n}\frac{1}{n^3}(i-1)^2= \displaystyle \lim_{x\to\infty}\frac{1}{n^3} \sum\limits_{n = 1}^{n-1}i^2$ This was from an solution to the...

S = 12-22+32-42...+20092 Attempt= S = (1+2)(1-2)+(3+4)(3-4)+...+(2007-2008)(2007+2008) [can we write this as -(1+2+3+4+5...2008) if yes, then why ?) +20092 Stuck after this.

Homework Statement y''+0.1y'+y=1+2\sum_{k=1}^{n}(-1)^{k}u_{k\pi}(t) and quiescent initial conditions. Homework Equations None. The Attempt at a Solution (s^{2}+0.1s+1)Y(s)=\mathcal{L}\{1\}+2\sum_{k=1}^{n}(-1)^{k}\mathcal{L}\big\{ u_{k\pi}(t)\big\} I'm not sure if this step was...

How can I do the summation of x^2 from 0 to 3 without the use of any built-in functions? I know a for loop is involved, but I can't get it to work.

Greetings, This is just an opinion question about notations. Having learned the basics of bra-ket notation and using the ESC, as far as I can tell, ESC is just plain better, at least when dealing with finite bases. Using bras and kets, you can represent and manipulate states using...

I've never seen this notation before. What does the ellipsis right after the first summation mean: \begin{equation} \label{aixi_eq} a_t^* = \arg\max\limits_{a_t}\sum\limits_{o_t r_t} \dots \max\limits_{a_{t+m}}\sum\limits_{o_{t+m} r_{t+m}}[r_t + \dots + r_{t+m}]...

Homework Statement Find the limit of the series \lim_{n \rightarrow \infty} \sum_{i=1}^{n} cos (i \theta / n) , 0≤θ≤π/2 Homework Equations The Attempt at a Solution I know that the expansion looks like \cos \theta / n + cos 2 \theta / n + ... + cos \theta , but I couldn't begin...

I immediately thought of induction, so that is what I used, but I can't seem to make any progress past a certain point.

Determining the commutation relation of operators -- Einstein summation notation Homework Statement Determine the commutator [L_i, C_j] . Homework Equations L_i = \epsilon_{ijk}r_j p_k C_i = \epsilon_{ijk}A_j B_k [L_i, A_j] = i \hbar \epsilon_{ijk} A_k [L_i, B_j] = i \hbar...

So, I realize the basic theory behind Einstein Summation Convention is that any repeated set of indices implicitly indicates a sum over those indices. However, what if an index is repeated three times? For example, my mathematics professor posted this problem: εijkajaj = ? As you can...

Homework Statement I am trying to follow a step in the textbook but I don't understand. var\left(\frac{1}{N}\sum_{n=0}^{N-1}w[n]\right)\\ =\frac{1}{N^2}\sum_{n=0}^{N-1}var(w[n]) where w[n] is a Gaussian random variable with mean = 0 and variance = 1 Homework Equations Var(X) =...

I understand the simplest application of the summation convention. x_{i}y_{i} I create a sum of terms such that in each term the subscripts are the same i.e. x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}+... But now when I look at understanding summation convention applied to the generalised Hooke's law...

Homework Statement Verify the following summations using Maple (see image). Homework Equations None The Attempt at a Solution For the first one, I enter sum(k^3, k=1..n); in Maple, and the result is 1/4*(n+1)^4-1/2*(n+1)^3+1/4*(n+1)^2, which is definitely not the...

Homework Statement 2n Ʃ (k) k=1 The Attempt at a Solution 2n Ʃ (k) = 2n(2n+1)/2 (This is just a shot in the dark.) k=1

I know how to do it normally, but this confuses me. Can you show me how to do it? And what were the steps you took... etc. Initially, I thought of putting them inside brtackets and work out the last one at the right first, but I'd need to know what K and J are, it's confusing. Can anyone make...

whats the summation of a^n for a>1 over summation n=0 to N thanks.

Hi! I'n my quantum mechanics homework I've been asked to proved the Poisson summation formula. The mathematicians seem to use abstract and confusing notation when proving this kind of thing so I'm hoping for some help from physicists in standard notation ;) I'm starting with a function f(x) =...

Folks, I am struggling to see what is happening here particularly when ## \displaystyle \sum_{i=1}^{n-1}## transforms into ##\displaystyle \int_{x_1^e}^{x_{n}^e}## ##\displaystyle 0=\sum_{i=1}^{n-1} \left [ \int_{x_i^e}^{x_{i+1}^e} (a \frac{dw}{dx} \frac{du}{dx}+cwu-wf )dx- \left [ w(x)...

Hello i want to know if we have a series with this equation and change it to this form the statement in Summation change or not?( this changes it right? where can i find the Summation notes?

Prove that the $\sum\limits_{k = 0}^n\cos k\theta = \text{Re}\left(\frac{1 - e^{i(n + 1)\theta}}{1 - e^{i\theta}}\right)$ simplifies to $$ \sum\limits_{k = 0}^n\cos k\theta = \frac{\sin\left(\frac{n + 1}{2}\theta\right)}{\sin\frac{\theta}{2}}\cos\frac{n}{2}\theta $$ So I have that the real part...

Just to be clear, I understand the difference between sigma summation and integration. Sigma summation is, put simply, the discrete version of integration. Rather than a continuous sum of a function for given values, sigma summation provides a sum of a function for given regions that is...

Homework Statement Hello! I'm guessing this is precalculus. There is an intermediate step in a simplifying process and I got to: \sum_{x=1}^n xA^{-Bx} Where A is a constant and B is a constant. Homework Equations I was wondering how to write this without the summation sign...

If I know that \sum_{k=1}^n a_{ik} = 1 and \sum_{j=1}^n b_{kj} = 1, why is the following permitted? \sum_{j=1}^n \sum_{k=1}^n a_{ik}b_{kj} = \left(\sum_{j=1}^n b_{kj}\right) \left(\sum_{k=1}^n a_{ik}\right) = 1\cdot 1 = 1 Thanks!

Homework Statement Here's my question. My school recently taught me finding summation using method of difference and what my teacher taught was just involving 2 partial fractions. But this question appeared in my exercise given by my teacher. r th term: (2r-1)/r(r+1)(r+2). Find summation...

I'm reading "An Introduction to Mathematical Reasoning," by Peter Eccles. It has some interesting exercises, and right now I'm stuck on this one: "Prove that \[\frac1n\sum_{i=1}^nx_i \geq \left(\prod_{i=1}^nx_i\right)^{1/n}\] for positive integers \(n\) and positive real numbers \(x_i\)."...

Homework Statement What is the sum of the following series? log(3/2)+log(4/3)+log(5/4)+...log(200/199). Where log(x) is log base 10 of x. Homework Equations The Attempt at a Solution Evidently, the previous form equals: log(3/2*4/3*5/4*...200/199) I'm missing something -...

Homework Statement I have attached the question, along with the solution in the picture attached. This is one of the few questions I have encountered that I completely have no idea what the solution is trying to do... It's like they do not make any sense at all! Confused by 1. The...

Homework Statement I want to change the order of triple summation. it follows: $$\sum^N_{k=0} f(k) \sum^k_{n=0}\sum^{N-k}_{m=0} g(k,n)h(k,m)A(n+m)$$ => I need to set the variable x(=n+m) go from 0 to N firstly, and then further go on... $$\sum^N_{x=0} A(x) \cdots \cdots$$ But, I...

Hi, I'm currently taking Discrete Mathematics and I'm working on a mathematical induction problem that's a little different than usual because it has a summation in it. What I basically want to know is did I do parts A and C correctly? Homework Statement Homework Equations The Attempt at a...

can you please tell me what is the basic difference between summation and integration..? i was going through the Poisson distribution function and in one case it was discrete and we had to make summation to get the result and other cases for continuous function we integrated it...now what is...