Sum Definition and 1000 Threads

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...

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

My book wants to sum the number of particles in each energy states for a gas of bosons, that it calculate the infinity sum: Ʃn(E) now if the E's are narrowly spaced it says we can approximate this with an integral. ∫n(E)ρ(E)dE Now can anyone tell me using the definition of the...

Homework Statement How to calculate? ## \sum _{i,j} \langle 0|\prod_n \hat{C}_n \hat{C}^+_i\hat{C}_j \prod_n \hat{C}^+_n|0 \rangle ##Homework Equations ##\hat{C}^+, \hat{C}## are fermionic operators. ##\{\hat{C}_i,\hat{C}^+_j\}=\delta_{i,j}##The Attempt at a Solution I have a question. What is...

I don't understand why the answer to this summation: [SIZE="1"]n [SIZE="3"]Ʃ 1/2 [SIZE="1"]i = 0 is (n+1)/2 Why isn't it just n/2?

Homework Statement Calculate the checksum for the following 3 bytes, showing all working without the use of a calculator: ##\displaystyle 01101101 ## ##\displaystyle 00100111 ## ##\displaystyle 00101101 ## ##\displaystyle \text{------------}## ##\displaystyle \text{------------}##...

Difference between 2 "Sum of n terms of geometric series" formulas Notation A) S_{n}= \sum_{k=0}^{n - 1} ar^{k} = ar^{0} + ar^{1} + ar^{2} +...+ ar^{n-1} = \frac{a(1-r^{n})}{1-r} Proof: S_{n}= ar^{0} + ar^{1} + ar^{2} +...+ ar^{n-1} - r*S_{n}= ar^{1} + ar^{2} + ar^{3} +...+ ar^{n}...

If $$x,\;y,\;z\in\mathbb{C}$$ and $$x+y+z=2$$, $$x^2+y^2+z^2=3$$, $$xyz=4$$, calculate $$\frac{1}{xy+z-1}+\frac{1}{yz+x-1}+\frac{1}{xz+y-1}$$.

Find the series sum ln2/2 – ln3/3 + ln4/4 – ln5/5 + ….

First, I've had to find the Fourier series of F(t) = |sin(t)|, which I've calculated as f(t) = \frac{2}{\pi} + \sum_{n=1}^{\infty}\frac{4cos(2nt)}{\pi-4\pi n^2} I'm pretty sure that's right, but now I need to evaluate the sum using the above Fourier series...

Hi everyone, I'm having some trouble with the concept of the direct sum and product of representations. Say I have two representations \rho_1 , \rho_2 of a group G on vector spaces V_1, V_2 respectively. Then I know their direct sum and their product are defined as \rho_1 \oplus \rho_2 : G...

a + b + c = a * b * c where a, b, c are positive integers. I can think of only one solution to this. {1, 2, 3}. Is there any other solution to it? Can you prove or disprove?

{Edit: as of 3:55 eastern time, made corrections to tex and itex mistakes}Is this all kosher in terms of demonstrating accuracy and comprehension of the notation {a_{1} + a_{2}...} = \lim_{n\rightarrow ∞ } \sum_{n=1}^{n} a_{n} So the lower case represents sequences and upper case represents...

I have a MySQL database and I have the problem that queries take way too much time. I want to optimize the database and one way would be to save data into a string, instead of in rows (the string would be replicates of the same condition). Each string is an interval of 4 seconds, to reduce the...

A Binomial distribution has a standard normal limiting distribution, i.e. (X-E[X])/se(X) -> N(0,1), where X is the sum of independent and identically distributed Bernoulli variables. Does this hold even when i) the Bernoulli variables are independent but non-identically distributed? That...

Homework Statement An impedance of 7-2jΩ and an impedance of 4+2jΩ and a current source of 97.5A are all connected in parallel. Find the individual branch currents. Homework Equations Current-divider rule: I1=I*Z1/(Z1+Z2) (only have to consider the magnitudes of the impedances, not the angle)...

Here is the question: Here is a link to the question: Help with precalculus! Sum or difference formula? - Yahoo! Answers I have posted a link there to this question so the OP can find my response.

Homework Statement Theorem: the numbers in the set {99, 999, 9999, ... } cannot be written as two squared integers, but at least one can be expressed as the sum of 3 squared integers. Homework Equations Well there are a lot of examples but let's go with 32 + 32 + 92 = 99 We may...

Homework Statement Find the sum using integration: lim_{n→∞} \frac{n}{(n+1)^2} + ... + \frac{n}{(2n)^2} Homework Equations The Attempt at a Solution I think this requires a clever construction of a series of an finite integral which after integration gives the series. Then it can...

Hi all, Saw this problem and was wondering if there was a simpler way to do this besides listing out the possible combinations. In a game, each token has one possible value: 1, 5, or 10. How many different combinations of these tokens will give us a total sum of 17?

"Sum to Product" Trigonometric identity does not work Hi, The identity [SIZE="3"]sin(u) + sin(v) = 2 * sin (\frac{u+v}{2}) * cos(\frac{u-v}{2}) http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities Does not always work. I put the equation...

Evaluate $$\frac{2^2-2}{2!}+\frac{3^2-2}{3!}+\frac{4^2-2}{4!}+\cdots+\frac{2012^2-2}{2012!}$$