Sum Definition and 1000 Threads

Please compute the following sum: S_n=\sum_{k=1}^{n}\frac{n!}{(k-1)!(n-k)!}

Is the following statement true? I am trying to see if I can use it as a lemma for a larger proof: Let ##V## be a vector space and let ##W, W_{1},W_{2}...W_{k} ## be subspaces of ##V##. Suppose that ## W_{1} \bigoplus W_{2} \bigoplus ... \bigoplus W_{k} = W ## Then is it always the case that...

Homework Statement I have no idea where the sum of x^2 comes from, from the information I posted. I know it must be something pretty simple but its completely going over my head. In the picture that I've attached, I am wondering where the 2431.72, 4901.66, and 3252.44 come from. Thank you...

Also would someone mind checking my work on these problems too? My answers are in BOLD 2a)Draw the truth table corresponding to $f$(X,Y,Z) = \piM(2,4,6) ANSWER: x y z | f 0 0 0| 1 0 0 1| 1 0 1 0| 0 0 1 1| 1 1 0 0| 0 1 0 1| 1 1 1 0| 0 1 1 1| 1 2b) Write out the canonical product of sums...

Homework Statement This is not a homework question, but I'm facing this from my research. I have N complex numbers defined as x_{n}=|\alpha_n| \cdot e^{j \theta_n} for n = 1,\ldots,N and my observation is the sum of those numbers r = \sum_{n=1}^{N} x_n . From the observation r, I...

Dear, I assume that a signal S is expressed as S = a*S1 + b*S2, where a, b are weight constant, and S1, S2 are the different signals. In addition, S1, S2 have similar distribution such as Gaussian or Laplacian distribution, and theirs pdf is p_S1 and p_S2. In the above...

Dear, I assume that a signal S is expressed as S = a*S1 + b*S2, where a, b are weight constant, and S1, S2 are the different signals. In addition, S1, S2 have similar distribution such as Gaussian or Laplacian distribution, and theirs pdf is p_S1 and p_S2. In the above assumption...

Homework Statement A signal E(t) is made up of three terms, each having the same frequency but differing in phase: E(t) = E0cos(ωt) + E0cos(ωt + δ) + E0cos(ωt + 2δ) It is possible to find the amplitude of the sum vector by summing each vector described as a magnitude multiplied by a...

\sum_{i=1}^{n}[i/2^i] Have looked and looked and cannot find it anywhere. EDITED: To correct mistake.

Is this true? I am studying direct sums and was wondering if the following statement holds? It seems to be true if one considers the proof of the dimension theorem, but I need to be sure, so I can steer my proof toward a particular direction. ## N(T) \bigoplus R(T) = V ## where ##V## is the...

This is a challenge problem I thought of: Given a real-valued matrix A, develop an algorithm that finds the submatrix with the greatest sum of its element. (If there's a tie, just return an arbitrary submatrix that's tied for the win.) Is there a way other than brute force?

I'm curious about whether a statement I conjecture about direct sums is true. Suppose that ##V## is a finite-dimensional vector space and ##W##,##W_{1}##,##W_{2}## are subspaces of ##V##. Let ## V = W_{1} \bigoplus W ## and ## V = W_{2} \bigoplus W ##. Then is it the case that ## W_{1} = W_{2}...

Homework Statement Compute the integral that is highlighted in MyWork.jpg using Riemann sums using left and right endpoints. Homework Equations ##x_i* = a + i Δx## ##*x_i = a + i Δx - Δx## ##Σ_{i=1}^{n} i = n(n+1)/2## ##Σ_{i=1}^{n} i^2 = n(n+1)(2n+1)/6## The Attempt at a Solution My...

We know a standard matsubara frequency sum that -\sum_{n}\frac{\xi}{2\pi ni-\xi}=n_B(\xi), but this looks contradictory to the well-known formula \sum_{n}\frac{\xi}{n^2+\xi^2}=\pi \coth(\pi \xi) if we take the imaginary part of the former sum. I know this matsubara frequency sum depends on...

Homework Statement If the product of the numbers R and 11/S is the same as their sum, find the value of S. Homework Equations N/A The Attempt at a Solution I am suspecting that the only set of 2 numbers that have the same sum and product is 2 and 2. So I guess R is 2, 11/S is...

Let $x$ be a complex number such that x^{2011}=1 and $x\ne1$. Compute the sum \frac{x^2}{x-1}+\frac{x^4}{x^2-1}+\frac{x^6}{x^3-1}+\cdots+\frac{x^{4020}}{x^{2010}-1}.

Finding Two Vectors from Given Linear Combination Homework Statement If v + w = (5,1) and v - w = (1,5), compute and draw v and w. Homework Equations v + w = (5,1) v - w = (1,5) The Attempt at a Solution I understand how to find the sum of two vectors, but I'm confused on how to find...

Problem: Through transformation with orthogonal matrix $O$, the problem \hat{b}=\underset{b}{\operatorname{arg min}}||y-Xb||^2 is equivalent to \hat{b}=\underset{b}{\operatorname{arg min}}||y^{*}-X^{*}b||^2, where $y$ and $y^{*}$ are in $\mathbb{R}^m$, $X$ and $X^{*}$ are in $\mathbb{R}^{m...

Find the exact value of the series \frac{1}{4!}+\frac{4!}{8!}+\frac{8!}{12!}+\frac{12!}{16!}+\cdots\cdots

Homework Statement I want to show that $$ \tan^{-1}(x)=\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}x^{2n+1}. $$ Homework Equations I start with $$ \int\frac{1}{1+x^{2}}dx. $$ The Attempt at a Solution I want to be able to do the following: $$...

Hello MHB, I have no good ideas on how to go about solving the following: Let $f:[a,b]\to\mathbb R$ and $g:[a,b]\to\mathbb R$ be real values functions both of which are differentiable in $(a,b)$. Show that there is an $x\in(a,b)$ such that...

Homework Statement Let \hat{u}_k the Fourier coefficients of 2-periodic function u(t)=t with t\in [0,2). Evaluate the sum of the serie: \sum_{k=-\infty}^{\infty}\hat{u}_k e^{\pi i k t} for t= 2 Ok, I think there is a trick that I don't know... \sum_{k=-\infty}^{\infty}\hat{u}_k...

Homework Statement Project Euler Problem 1. If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23. Find the sum of all the multiples of 3 or 5 below 1000. Hello, there I do understand what a multiple is, and...

Find \sum_{n=0}^\infty \frac{\cos(nx)}{2^n}.

If $a, b$ and $c$ are three real numbers such that $|a-b|\ge|c|$, $|b-c|\ge|a|$ and $|c-a|\ge|b|$, then prove that one of $a, b$ or $c$ is the sum of the other two.

Dear friends, Over the past week, I tried to plot implicit function by mathematica but failed. I am very disappointed. Hopefully someone help me at this time. My equation is given by (see below figure): Where z0 := 6 d := 12 k := 11800 w0 := 0.025 w[z_] := sqrt[w0^2*(1 + (z/z0)^2)]...

Hi all, For finding average we take the sum of sequence numbers and divide by the number of elements. Why for variance this changes to number of elements minus 1. -Devanand T

I don't understand any of the steps that were made in the book. So I will try and solve it on my own so please let me know where I am going wrong. The problem is in the paint document. The general Term for the infinite series given is n(n+1) where n is greater than one and integral. LHS we...

Here is the question: I have posted a link there to this topic so the OP can see my work.

I am following Friedberg's text and having some trouble understanding some of the theorems regarding diagonalizability. The proofs seem to skip some steps, so I guess I need to work through them a bit more slowly. Given a linear operator ## T:V → V ##, with eigenspaces ## \{ E_{...

Hello everybody, I'm trying to understand some steps in the evolution of calculus, and in a .pdf found in the internet I read the document: http://www.ugr.es/~mmartins/old_web/Docencia/Old/Docencia-Matematicas/Historia_de_la_matematica/clase_3-web.pdf , in pags. 14-15. I want to solve the to...

I'm self-studying Linear Algebra and the book I'm using is Linear Algebra done right by Sheldon Axler but I came across something that I don't understand .- Suppose \mathrm U is the set of all elements of \mathbb F ^3 whose second and third coordinates equal 0, and \mathrm W is the set of all...

We have for two random variables X and Y (one sum runs over j and one over k): E(X+Y) = ƩƩ(xj+yk)P(X=xk,Y=yk) = ƩƩxjP(X=xk,Y=yk) + ƩƩykP(X=xk,Y=yk) Now this can be simplified to obtain E(X+Y)=E(X)+E(Y) if we use that: P(X=xk,Y=yk) = P(X=xk)P(Y=yk), because then (and same goes the other...

Homework Statement Consider the series Ʃ 1/[k(k+2)]; n=1 to infinity Find the formula for the partial sum Sn 2. The attempt at a solution I have calculated the first 5 terms of the sequence as follows, but I can't see any pattern. Am I doing this right? S1=1/3 S2=1/3+1/8=11/24...

$ X=3^{10000}$ Given $log_{10} 3=0.4771$ All the digits sum of X =A All the digits sum of A =B All the digits sum of B =C please find C

In an earlier post i was shown how to represent an integral as an infinite sum. So why is the anti derivative a summation by definition? For example, the derivative dy/dx is found by f(x+h)-f(x)/h.

This thread will be dedicated for a trial to prove the following \sum_{k\geq 1} \frac{H^2_k}{k^2}=\frac{17}{4}\zeta(4)=\frac{17\pi^4}{360} \mbox{where }\,\,H^2_k =\left( 1+\frac{1}{2}+\frac{1}{3}+\cdots \frac{1}{k}\right)^2 In this paper the authors give solutions to the sum and others ...

Prove the following \sum_{k\geq 1} \frac{H^2_k}{k^2}=\frac{17}{4}\zeta(4)=\frac{17\pi^4}{360} \mbox{where }\,\,H^2_k =\left( 1+\frac{1}{2}+\frac{1}{3}+\cdots \frac{1}{k}\right)^2

The real numbers x and y satisfy x^3-3x^2+5x-17=0 and y^3-3y^2+5y+11=0. Determine the value of x+y.

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.

Problem: The sum of two real numbers is 1. What is the minimum value of the sum of the squares of the two numbers? I have already managed to solve the problem algebraically (by substitution and completing-the-square we arrive at a minimum value of 0.5), but what I am interested in is a...

I was wondering if someone could take the time to show a proof for the sum and difference identity for sine. I've seen and learned to understand some other identities, but never this one. I've been trying to understand more of the "why" than the "how" of mathematics, and this one is very...

I've looked over this peculiar problem and the many different forums that have tried to resolve it yet I'm confused as to why the answer simply isn't zero. for those not familiar with this problem, it is this Ʃ of (-1)^n as n goes from 0=∞ which...

Homework Statement Prove that every integer >17 can be written as the sum of 3 integers >1 that are pairwise relatively prime. The Attempt at a Solution I already proved the case for even integers. Now I am just working on the case for odd integers. I know that it has to be the sum of 3...

Homework Statement Prove that every integer bigger than 6 can be written as a sum of 2 integers bigger than 1 which are relatively prime. The Attempt at a Solution Ill first look at the case where our number is odd. Let x be an odd integer. I will just add (x-2)+2=x since x is odd so is x-2...

If we are given with set of n natural numbers and asked to find the sum of all possible products of two number.. E.G: {1,2,3} is given then , 1*1+1*2+1*3+2*2+2*3+3*3=s how to find the general formula of SUM for given n-tuple containing consecutive natural numbers not necessarily starts with 1 or...

how to find the sum, 1^4+2^4...n^4 i tried myself but couldn't find an approach , when i googled it ifound the formula but not proof

Homework Statement (a) Suppose you have two arrows of equal length on a tabletop. If you can move them to point in any direction but they must remain on the tabletop, how many distinct patterns are possible such that the arrows, treated as vectors, sum to...

Let $a,b,c$ be positive real numbers with sum $3$. Prove that $√a+√b+√c≥ab+bc+ca$.

Find \sum_{x=0}^{101}\frac{\frac{2x}{101}-1}{\frac{3x^2}{10201}-\frac{3x}{101}+1}.