Summation Definition and 610 Threads

Edit: LOTS OF TYPOS (sorry guys) Let: f(r) = e^{-(a-r)^2} g(r) = r e^{-(a-r)^2} Where a is some constant Can: \dfrac{ \sum\limits^{r=\infty}_{r=-\infty} g(r) } {\sum\limits^{r=\infty}_{r=-\infty} f(r) } Be simplified?

Hello all, I'm aware of the Monte Carlo Summation method in discrete spaces, where you can approximate a very long summation over the entire space by a shorter one with only a few randomly selected terms from the original summation (weighted by the inverse probability density of them being...

Is there a general formula for something like \sum_{n=0}^{\infty} \left( f(n) \times g(n) \right) For example, what is \sum_{n=0}^{\infty} \left( 3^n \times \frac{n!}{n^2} \right)

Homework Statement What is the sum of: Homework Equations N/AThe Attempt at a Solution I'm unsure how to start. Note: I'm in Grade 10, so I may not have the mathematical skills necessary to understand the solutions you provide. Any help/guidance would be appreciated.

How to evaluate the following expression? \sum_{i=0}^{N} \binom{N}{i} \left(-1\right)^{i}\left(\frac{1}{2+i}\right)^{k} regards, Bincy

Homework Statement Someone in school was showing me this proof or problem that, I believe, proves or yields π via this limit: Lim x-->0 of \frac{xπcot(πx)}{x}-\frac{1}{x} = tan(0) = 0 And that this somehow related to a summation \sum1/k^{2} as the sum goes from 1 to ∞. I don't...

So basically here's the deal: I believe there exists a P(x) defined on [-2π, 2π] such that over that interval P(x) = \sum^{\infty}_{n=0}[sin(πnx)] Its weird but I have a feeling that this might converge to a function such as tangent

Homework Statement For formatting sake I've copied a picture of the problem and attached it here: http://i.imgur.com/kOjTy.png Im not worried about the coding part right now I feel I can handle that, my main issue is trying to understand how the values in the summation are derived. It...

The sum of 1 + 2 + 3...n = n(n+1) / 2 - highest power term is n^2 sum of 1^2 + 2^2 + 3^2...n^2 - n(n+1)(2n+1) / 6 - highest power term is n^3 sum of 1^3 + 2^3 + 3^3...n^3 - it has highest power term of n^4 similarly 1^k +2^k ...n^k - it has highest power term of n^(k+1) Is it a coincidence...

[FONT=verdana]Find the sum of n terms: 1+2(1-a) +3(1-a)(1-2a)...k(1-a)(1-2a)...\{1-(k-1)a\}

Show that $\displaystyle \sum_{n=0}^{\infty} (-1)^{n} \arctan \left( \frac{1}{2n+1} \right) = \arctan \Bigg( \text{tanh} \Big( \frac{\pi}{4} \Big) \Bigg)$.I'm tempted to give a hint (or two) right off the bat. But I'll wait.

Homework Statement http://desmond.imageshack.us/Himg100/scaled.php?server=100&filename=img20120327195119.jpg&res=medium Homework Equations The Attempt at a Solution I just plugged in ∞ for n [2+\frac{3}{∞}]2 (\frac{3}{∞}) = [2+0]2 (0) = 0Did I do the problem correctly? I might need a...

Hi Is it correct of me to say that I want to carry out the sum \sum_i{v_iw_i} where i\in\{x,y,z\}? Or is it most correct to say that i=\{x,y,z\}?Niles.

Hello, I am looking at a derivation that involves (note x is a column vector) \frac {d(\vec{x}^T\vec{x})} {d\vec{x}} = \vec{x}^{T} So I convert to summation notation and evaluate as follows \sum_{i,j} \frac {d(x_{i}x^{i})} {dx^{j}} \sum_{i,j} \frac {dx_{i}} {dx^{j}} x^{i} + \sum_{i,j}...

Let $k$ be a positive integer. Let $n=2^{k-1}$. Prove that, from $2n-1$ positive integers, one can select $n$ integers, such that their sum is divisible by $n$.

Homework Statement \frac{∂x_{i}}{∂x_{j}} = δ_{ij} Homework Equations \vec{r} = x_{i}e_{i} The Attempt at a Solution \frac{∂x_{i}}{∂x_{j}} = 1 iff i=j δ_{ij} = 1 iff i=j therefore \frac{∂x_{i}}{∂x_{j}} = δ_{ij} Homework Statement r^{2} = x_{k}x_{k} Homework...

Homework Statement I want to justify that \int_{0}^{1} \frac{f(x)}{1-x} \ dx = \int_{0}^{1} f(x) \sum_{k=0}^{\infty} x^{n} \ dx = \sum_{k=0}^{\infty} \int_{0}^{1} f(x) x^{n} \ dx Homework Equations The Attempt at a Solution I always thought changing the order of summation...

I have the following summation and I'm attempting to remove the summation notation. It appears to be the sum of a geometric series but I'm having a great deal of trouble with it. X is an unknown constant. $$\sum\limits_{i=2}^n (n - (n-i))x^{n-i}$$ Thanks.

How can I make this mathematically correct? I hope you see what I'm trying to do?... If you have a graph where: W=\displaystyle\int^{\theta_{2\pi}}_{\theta_{0}} \tau d\theta Then the estimated area with the trapesium rule: \displaystyle\int^{\theta_{2\pi}}_{\theta_{0}} \tau...

Homework Statement I have trouble with the summation notation. \sum_{i=0}^{k}\binom{k}{i}f_{n+i} How do I write this as a sequence based on the definition of Fibonacci sequence? Homework Equations Definition: f(0)=0 f(1)=1 f(n)=f(n-1) + f(n-2) for n>=2 Example: f(2) = f(1) +...

Determine a formula for the sum of \sum iri-1 in terms of n and r. I am stuck on this, i don't understand what to do with the "i" infront of the ri-1i know that \sum ri-1 = (1-rn ) /1-r all sums are for an index of i=1 to n I...

I am trying to understand summation notation and there are a few inconsistencies in my head that I would like to clear up. Suppose C is an m*n matrix and \vec{x} is a 1*m row vector. Then, \vec{x}C = \sum_{i} x_{i}C_{ij} = \sum_{i} C_{ij}x_{i} = \sum_{i} {C_{ji}}^Tx_{i} = C^T \vec{x}...

u is a complex. Can this sum be simplified? If so, how? $\displaystyle \sum_{n}^{\infty}\frac{2u+5}{2}\left(-\frac{u}{2}\right)^n $ Thanks.

I have been working on representing the powers of numbers as a summation. This is as far as I have gotten. Power: 2 m^2 = \sum_{n=1}^m \left(2n -1\right) Power: 3 m^3 = \sum_{n=1}^m \left(3n^2 -3n +1\right) Power: 4 m^4 = \sum_{n=2}^m \left[6*(4n-6) * \left(\sum_{a=1}^{m-n+1}...

Hi, I am reading a paper, and at some point the authors claim that: \sum_{m=1}^{L+1}\frac{\prod_{\substack{l=1\\l\neq m}}^{L+1}\frac{\lambda(m)}{\lambda(m)-\lambda(l)}}{\lambda^r(m)}=0 the question is HOW? Any tiny hint will be highly appreciated. Thanks

How would I write 1 + n + n(n - 1) + n(n - 1)(n - 2) + ･･･ + n! in sigma notation? If it's possible.

Homework Statement Consider any sequence a1, a2,..., an and the nxn array of values bij = aiaj. Which terms in the array are involved in the sums L = Ʃ(between i=1 and n)Ʃ(between j=1 and i) bij and U = Ʃ(between j=1 and n)Ʃ(between i=1 and j) bij? Also, by symmetry, show that L=U. Homework...

Homework Statement I have the following function, f(t) = 2048 + 700cos(31.25*2\pi t) - 1100sin(125*2 \pi t) and I need to find 48 data points spread evenly between one peroid of the waveform. How would I go about doing this? Homework Equations The Attempt at a Solution...

Prove of summation claim ?? Hi every one,, any idea how to prove the following claim \sum_{i=0}^{n}a_iz^i=(1-z)^{\binom{m}{2}}(1+z)^{\binom{m+1}{2}} i think we need to use some derivatives, may be the second derivative will help. please help.

Hello everybody, I have some problems in finding an analytical expression for this product: \sum_{j=0}^{N}(N-j)e^{-ijy}\cdot\sum_{k=0}^{N}(N-k)e^{iky} . I have solved the problem for several Ns, applying the Euler rule 2\cos(x) = e^{ix} + e^{-ix} Now, I'm trying to express the...

I have just started on a course in Tensor calculus and I'm absolutely new to it, so I read that according to the summation convention, if an index appears twice, it means that the expression is summed over that index, but if it appears more than twice then the expression is meaningless. I want...

This problem came up in a project I'm doing for work, and I don't have a very extensive math background so I don't know how to solve it. I would appreciate any help you guys could give me. X1 is a constant Y1 is a constant Xn = aXn-1 + bYn-1 Yn = cXn-1 + dYn-1 For all n, for some constants...

Hi. I've finished my undergraduate math methods courses. Many times we had problems where we had a summation and an integral both acting on the same term, and we'd switch the order of the two operations without thinking about it. The professor would always say, "I can interchange these two...

Good day all, I received this picture through the facebook network last night. I took it as valid at first sight, but for some reason it bothered me; in other words, my skepticism kicked in. The author mentions and assumes the limit as "n goes to infinity" which is annotated in the...

It's been a while since I took Calc 2 and I am in a Linear Systems and Signals class right now. I'm looking at a solution on how to obtain a zero state response of a discrete time signal, but performing the summation confuses me. Can someone explain the steps they did? This is part A...

I am looking for a bound for the following expression S=\sum_{n=1}^N n^k e^{-an} where a>0 and k=1, 2, 3, or 4, apart from the obvious one: S\le \frac{n+1}{2} \sum_{n=1}^N e^{-an} = \frac{n+1}{2} \frac{1-e^{-Na}}{e^a-1}

Hy everyone! I want to include this two lines: m,n m+n=k but only in the bottom of the sumatory. Is that possible? Thanks in advance for the help!

Homework Statement The Spivak's Calculus Answer Book (3ed) states that, on page 17, \sum_{i \neq j} (x_{i}^{2}y_{j}^2 - x_{i}y_{i}x_{j}y_{j}) = 2\sum_{i < j}(x_{i}^{2}y_{j}^2 + x_{j}^{2}y_{i}^2 - x_{i}y_{i}x_{j}y_{j}) But as I speculate, I've got the following: \sum_{i \neq j}...

My logic is flawed somewhere, but I can't figure out where or why. So I've been playing with summation a bit and figured out a way to make equations for Ʃ^{n}_{k=1}K and Ʃ^{n}_{k=1}K^{2} That looks odd, so I'll just use Ʃ from now on, but realize that it is always from k=1 to n. ƩK is a series...

Homework Statement I need some advice on prooving this formula (f is an arbitrary function): \sum^{N}_{t=1}\sum^{N}_{s=1}f(t-s)=\sum^{N-1}_{τ=-Ν+1}(N-|τ|)f(τ) Thanks in advance

Does anybody know how to solve the next summation? or process, or mathematical program that can solve this? [PLAIN]http://img341.imageshack.us/img341/7939/unledkn.jpg

Homework Statement Show that f(x) = \sum_{i=1}^{\infty}\frac{2^{i}x - \lfloor 2^{i}x \rfloor}{2^{i}} is continuous at all real numbers, excluding integers. The Attempt at a Solution I've tried going about via |f(x) - f(y)| < ε, but am having trouble with this, since first, I don't get anywhere...

Homework Statement A line of buckets numbered 0,1,2... extends indefinitely to the left with an elephant behind each bucket. Initially, all of the buckets are empty but then peanuts start falling into bucket 0 at a rate of one per second for 2^12 seconds. Whenever 5 peanuts accumulate in a...

hi guys, new user, long time lurker. the following simple proof is proposed with the highlighted summations. you do not need to know what the proof is of to answer my question (it is that the payoff for a skew symmetric game, rock paper scissors, is zero). i need help understanding how you...

My understanding of the Einstein Summation convention is that you sum over the repeated indices. But when I look at the metric tensor for a flat space I know that g^{λ}_{λ} = 1 But the summation convention makes me think that it should equal the trace of the matrix g_{μσ}. So it should...

Homework Statement Hi I was trying to understand an algorithm analysis problem and I came to this point: [PLAIN]http://img820.imageshack.us/img820/1834/unledytw.png can someone explain me what he did there? Is there any other step between that should have been written in order for...

How can I show that \sum_{i=1}^n\;\frac1{i(i+1)}=\frac{n}{n+1} I've already figured out i can write it as \sum_{i=1}^n\;\frac1{i}-\sum_{i=1}^n\;\frac1{i+1} but as I'm a little drunk I can't figure out how to get from there to the formula. Sorry if I put this in the wrong sextion, but...

I'm trying to figure out what the summation notation of (1-x1)^θ * (1-x2)^θ * ...(1-xn)^θ would be for summation I know I need to convert this to a summation notation in order to solve my problem, but I can't figure out how to convert it. Any help will be appreciated.

Hi all, \sum_{i=1}^n k_i \Pi_{i=1}^n O_i(\mu) How to interpret this equation.

Why is the Gauss summation formula for complex parameters a,b,c: \displaystyle _2 F_1 (a,b;c;1) = \frac{\Gamma (c) \Gamma (c-a-b)}{\Gamma (c-a) \Gamma (c-b)} only valid for \text{Re}(c-a-b)>0,\;c\neq 0,-1,-2,-3,...?