Summation Definition and 610 Threads

Homework Statement My question is regarding a single step in a solution to a given problem. The step begins at: ##\large \frac{\partial \alpha _j}{\partial x ^i} \frac{\partial x^i}{y^p} \frac{\partial x^j}{\partial y^q} - \frac{\partial \alpha _j}{\partial x ^i} \frac{\partial x^i}{\partial...

I have a function f(x,y) which i have defined in this way: a vector x and a vector y meshgrid[x,y] z= f(meshgrid[x,y]). how do i do a 2-d Fourier transform of f(x,y)? the transform must be done without using operations like fft, and must be done using summations written in the code.

Hi, I'm working with series solutions of differential equations and I have come across something that has troubled me other courses as well. given that \begin{equation} \sum_{n=0}^{\infty} c_{n+2}x^n+e^{-x} \sum_{n=0}^{\infty}c_{n}x^n \\ \text{where}\\...

I feel like Ramanujan Summation is just very bizarre. How can 1+2+3+4...=-1/12? It all rests in the assumption that ∑n=0∞(-1)n=.5. However, in calculus, limn→∞(-1)n=undefined. The limit does not exist. It is not 0, the average of -1 and 1 which are the only values of the function (if the domain...

Hello, in my QM class we arrived at the expression ##\langle \hat{H} \rangle = \Sigma_{even n} |C_n|^2 E_n = \frac{24}{n^2 \pi^2} \frac{\hbar^2}{2m} \frac{n^2 \pi^2}{L^2}##. The n terms cancel and we are left with ##\langle \hat{H} \rangle = \frac{12 \hbar^2}{mL^2} \Sigma_{even n} 1##. My...

Hello. I'm new to this forum. I'm starting a PhD – it's going to be a big long journey through the jungle that is CFD. I would like to arm myself with some tools before entering. The machete is Cartesian Tensors. I know the rules regarding free suffix's and dummy suffixes, but I'm having...

HI everyone, Imagine we are sampling of a gaussian signal along time and need to know the power/variance associated with the first N spectral components. So we take our favorite fft algorithm to get the PSD. The error associated with a given estimated spectral component f(w) (w is the...

I've been reading a bit about the very intriguing summation \displaystyle \sum_{n=0}^{\infty} {n} and it seems \frac{-1}{12} is the result but apparently with a lot of subtleties and caveats. It is those that I am trying to understand now. At first reading it appeared totally incongruous to...

I know n(n+1)/2 solves from 1... n by 1's Is there a formula where you can add up by 3's? Example: 3 + 6 + 9 + 12 ... n

Prove the following: $$\sum_{k=n}^{2n-2}\frac{1}{k^2}<\frac{1}{2n-1}$$ where $$2\le n$$

Homework Statement It is very well known that ## \sum^{\infty}_{n=0}x^n=\frac{1}{1-x}##. How to show that ## \sum^{\infty}_{n=0}\frac{(a)_n}{n!}x^n=\frac{1}{(1-x)^a}## Where ##(a)_n=\frac{\Gamma(a+n)}{\Gamma(a)}## [/B]Homework Equations ## \Gamma(x)=\int^{\infty}_0 e^{-t}t^{x-1}dt## The...

Homework Statement Find \sum\limits_{k=0}^{n}k^2{n\choose k}(\frac{1}{3})^k(\frac{2}{3})^{n-k} Homework Equations -Binomial theorem The Attempt at a Solution I am using the binomial coefficient identity {n\choose k}=\frac{n}{k}{{n-1}\choose {k-1}}: \sum\limits_{k=0}^{n}k^2{n\choose...

Homework Statement The average number of mRNAs in the cell at any time t is <m>(t) = Σ m * p(t). Sum over all the differential equations derived in a) in order to obtain a differential equation for <m>(t) Homework Equations So the differential equation I got in a) was dp/dt = (-kp * Pm) - (m *...

Homework Statement *This is not my whole problem, I am only stuck on how to interpret one part of the question. Put simply, I want to find the expression for the density of energy levels in a given energy band per unit volume (in some crystal structure). Say I have an infinitesimal interval of...

I tried posting this question in this forum a couple of weeks ago, but didn't get an answer to my question. I'm going to try posting it again using the formatting template so that it is hopefully clearer. I am also not sure if this is the right forum to be posting this in. It is a problem I ran...

Hi, I am trying to find the error propagated by calculating the sum of a set of mass flow rates collected over the same length of time. The sum of mass flow rates can be calculated with two approaches, since the collection time is the same for all of them. Approach (1) is adding up all of the...

So I have a question I am hoping someone can help me answer. I am trying to compute transfer functions for a hammer impact with an accelerometer response on a cylinder. Please see the attached photo. http://imgur.com/F8DGwl2For some reason the picture did not attach but I have uploaded to...

Would it be possible to write an equation utilizing a summation of a modular function of a Cartesian function, whose degree is dependent upon the index of the root, in that it predicts the digits less than 1 of the root, that when summed equals the computed value sqrt( n )? I already have what...

Suppose we have two laser diodes that are made to transmit light at the same wavelength and intensity. Also, suppose that we place them in an open space and separate them by a distance such that when their generated beams intersect at one point in space and one point only. Further suppose that...

Hello, I can not find the way to solve the following equation: sum of k^2 f(k) from k=1 to n. In Particular, k^2 * (1/k)

This is not only a question strictly about mathematics, but in science or any other quantitative field in which there is an integration - or a summation that is like a discrete integration. [ A ] the parameter that is considered the input variable for the integration/summati - i.e., the x of dx...

I was wondering if anyone knew the standard notation for the following. Suppose I have functions ##f_1,f_2 \ldots,f_n##, is there a compact way of writing ##f_1 \circ f_2 \circ \ldots \circ f_n## ? I am debating whether ##\bigcirc^n_{i=1} f_i## is proper or good notation. Have anyone encountered...

Homework Statement A proton is composed of three quarks: two "up" quarks, each having charge +2e/3, and one "down" quark, having charge -e/3. Suppose that the three quarks are equidistant from one another. Take the distance to be 3×10-15 m and calculate the potential energy of the subsystem of...

Homework Statement $$ \sum_{n=1}^\infty\frac{1}{1+(a+nb)^2} = ? $$ 2. The attempt at a solution I approximated the result by integration, $$ \begin{align} \sum_{n=1}^\infty \frac{1}{1+(a+nb)^2} &\approx \lim_{N \rightarrow +\infty} {\int_{0}^N \frac{1}{1+(a+bx)^2} dx}\\ &= \lim_{N...

Homework Statement Prove that $$\sum_{r=1}^{b-1}[\frac{ra}{b}]=\frac{(a-1)(b-1)}{2}$$ where [.] denotes greatest integer function and a & b have no common factors. Homework Equations ##n\le [n]<n+1## <x> denotes fractional part of x. 3. The Attempt at a Solution I first added and subtracted...

How does e-Δ2/δ2 ≈ 1-Δ2/δ2 When Δ<<δ ? I'm sure it's a basic summation I'm unaware of.

Homework Statement This isn't really a problem I've been given, but questions i have about how the author of my textbook, Leon Couch, Digital and Analog communications Systems, found the PSD (power spectral density) of an digital NRZ pulse train. Homework Equations The PSD of a periodic signal...

In this problem, Spivak shows how to derive formulas to summations. They start by showing the method for 1^2 + 2^2 + ... + n^2 as follows: (k + 1)^3 - k^3 = 3k^2 + 3k + 1 Writing this formula for k = 1, 2, ..., n and adding, we obtain 2^3 - 1^3 = 3*1^2 + 3*1 + 1 3^3 - 2^3 = 3*2^2 + 3*2 + 1 ...

what is the result for the following double summation: ##\sum\limits_{i \neq j}^{\infty}\alpha^i\alpha^j ## where ## i, j =0,1,2,...##

I'm interested in the following inequality (which may or may not be true) Theorem 1: ##( \sum_{i=1}^n \frac{a_i} {n}\ )( \sum_{i=1}^n \frac{1} {b_i}\ ) > \sum_{i=1}^n \frac{a_i} {b_i}\ ## Where ##n ≥ 2, a_1 < a_2 < ... < a_n## and ##b_1 < b_2 < ... < b_n##. My attempt at a proof: 1) When n =...

Homework Statement By considering ∑z2n-1, where z=eiθ, show that Σcos(2n-1)θ=sin(2Nθ)/2sinθ. (Σ means summation from 1 to N)Homework Equations Just a guess. S=a(1-r^n)/(1-r) The Attempt at a Solution I was thinking this but it doesn't seem to work very well...

Hi, I've enclosed my problem and attempt at solution below. I had problems with the latex so I photographed a picture of my work. The first top half is my attempt at the solution and the bottom is the solution that Spivak provides. I don't understand how he reached his solution and I don't...

Homework Statement ##\sum\limits_{r=0}^n\frac{1}{^nC_r}=a##. Then find the value of $$\sum\sum\limits_{0\le i<j\le n}(\frac{i}{^nC_i}+\frac{j}{^nC_j})$$ Homework Equations I have used two equations which I derived myself. This is the first one. The second one is: 3. The Attempt at a...

Hi. I'm sorry to bother you, but I was trying to find the symbol used for a vector of observations that doesn't implicitly infre multiplication or summation. I'm trying to express an inequality at the simple and general levels so that \muAa \ne \muAA, \muaa The idea is that this inequality...

Homework Statement If ##|P(x)|<=|e^{x-1}-1|## for all x> 0 where ##P(x)=\sum\limits_{r=0}^na_rx^r## then prove that ##|\sum\limits_{r=0}^nra_r|<=1## Homework Equations None The Attempt at a Solution ##P(1)=a_0+a_1+...## If the constants are positive, then ##P(1)<=|e^0-1|## So P(1)<=0 so...

Homework Statement ∞ Σ ( x^n/ln(n+5) ) n=1 find the value of x that the above series converges Homework EquationsThe Attempt at a Solution i cal. it by ratio test and i found that |x|<1 but when i input (-1,1) into my webwork...it said it's wrong[/B]

I need to prove that the following summation is true: ∑(-1)^n * cos(nx) / n^2 = (3x^2 - π^2)/12 How would I tackle this problem?

I found the following identity in a paper: ## \sum_{l=1}^{\infty}exp(-\pi\alpha l^2)=(\frac{1}{2\sqrt{\alpha}}-\frac{1}{2})+\frac{1}{\sqrt{\alpha}}\sum_{l=1}^{\infty}exp(\frac{-\pi l^2}{\alpha}) ## Someone please let me give some hints on how to prove this.

Hi, I need to plot the last function of this: But I don't know how to generate the sum. I know the for loop is totally wrong, but I can't go any further. This is what I have: Can someone fix the summation loop part for me? Thanks in advance

Homework Statement determine series convergence of divergence summation (n=1 to infinity) n/n^2 +1 Homework EquationsThe Attempt at a Solution I take the limit comparison limit (1/n)/ (n/(n^2 +1) =1 for 1/n if i use p series the series diverge if i use the method to take limit of sequence...

I have been looking through some notes on fermion wavefunction operators and noticed some summations involving indexes repeated 3 times.I know this is not allowed when using the Einstein summation convention. So my question is : is the Einstein convention not used in Quantum mechanics ? and do...

Hi, I'm looking for a program that spits out fully summed index equations. For example T_{ii} in, out comes T_{11}+T_{22}+... and so on, with Einstein summation convention.

Homework Statement A and B are matrices and x is a position vector. Show that $$\sum_{v=1}^n A_{\mu v}(\sum_{\alpha = 1}^n B_{v\alpha}x_{\alpha})=\sum_{v=1}^n \sum_{\alpha = 1}^n (A_{\mu v} B_{v\alpha}x_{\alpha})$$ $$= \sum_{\alpha = 1}^n \sum_{v=1}^n(A_{\mu v} B_{v\alpha}x_{\alpha})$$ $$=...

Homework Statement The sum we are given is Σ(from x=0->∞) [(x^2)(2^x)]/x!. We are asked to find the exact value of this sum using concepts discussed in class which include poisson random variables, and their expected values. The Attempt at a Solution [/B] So i know the solution to the...

Homework Statement I have the following equation Aab= c ua ub Where Aab is a rank 2 tensor and ua is a vector and c is a scalar and a,b = {0,1,2,3}. I know both Aab , ua and ua I want to find c explicitly but I don't know how to interpret or calculate c = Aab / ( ua ub ) Does anyone...

Homework Statement Dear Mentors and PF helpers, Here's my question, I see these on my textbook but couldn't really understand how they derived this short cut. Please show me how they got to these. Thank you for your time. Homework Equations These is what I understand from now. The...

I encountered this expression while trying o express $f(x)=\ln\left({\frac{1+x}{1-x}}\right)$ in terms of a power series: $$\int \left[ \sum_{n=0}^{\infty}(-1)^n x^n + \sum_{n=0}^{\infty}x^n\right] \,dx$$ The book simplifies this expression as $\int \sum_{n=0}^{\infty}2x^{2n} \,dx$ by expanding...

Hi, I was trying to form a summation for ##y_1## and have provided a solution but do not quite understand how it was formulated. I was trying to look for general patters and besides a ##(-1)^{n+1}x^2n## in the numerator, I'm a little lost on how to find a general term for the denominator. Also...

A=limn→∞Rn=limn→∞[f(x1)Δx+f(x2)Δx+...+f(xn)Δx] Consider the function f(x)=4√x, 1≤x≤16. Using the above definition, determine which of the following expressions represents the area under the graph of f as a limit. I knew the correct answer was \sum \frac{15}{n} (4√x+\frac{15i}{n}) I figured...

Hello, Suppose we have: $$\begin{align} \sum_{n=1}^{\infty}\frac{1}{9n^2 + 3n - 2} &=\frac{1}{3}\sum_{n=1}^{\infty}\left(\frac{1}{3n - 1}-\frac{1}{3n + 2}\right)\\\\ &=\frac{1}{3}\sum_{n=1}^{\infty}\int_0^1\left(x^{3n-2}-x^{3n+1}\right){\rm d}x\\\\...