Sum Definition and 1000 Threads

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: n Ʃ 1/2 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)...

Homework Statement Note: I need help with part (c). Consider the representation P: S_3 \rightarrow GL_3 where P_{\sigma} is the permutation matrix associated to \sigma. a) Determine the character \chi_P : S_3 \rightarrow \mathbb{C} b) Find all the irreducible representations of S_3. c)...

Assume n\in\mathbb{N} and r\in\mathbb{R} are some fixed constants and r>0. I want to find some nice lower bound for the amount of elements (k_1,k_2,\ldots,k_n)\in\mathbb{N}^n such that 2^{k_1}+\cdots + 2^{k_n}\leq r. In other words, if we define a function f:\mathbb{N}^n\to\mathbb{R},\quad...

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 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 : (sin(u)...

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

Can someone please show me how the formula \sum^{∞}_{0}\frac{u^{x}}{x!} = e^{u} Is derived? Or link me to an explanation. Thanks! http://www.wolframalpha.com/input/?i=sum+of+%28%28a%5Ex%29%2F%28%28x%29%21%29%29+from+x%3D0+to+x+%3D+inf (Just to show you what I'm talking about)

Here is the question: Here is a link to the question: (CALCULUS)The sum of two numbers is k. Show that the sum of their squares is AT LEAST (1/2)(k^2)? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

I asked the following question on facebook Prove that \frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4} \, \cdot\cdot \cdot \,+ \frac{1}{n(n+1)}=\frac{n}{n+1}

Sum from 0 to infinity of [3/[(n+1)(5^n)]](x-3)^n HELP :) 1. ∞ Ʃ \frac{3}{(n+1)(5^{n}}*(x-3)n n=0 2. The first question I had to answer was: What is f(3)? I found the first 4 terms to be: 3, 3/10(x-3), 3/75(x-3)2, 3/189(x-3)3 So f(3) equals 3, I'm pretty sure. Because...

B(t) is a standard Brownian Motion. u and v are both => 0. What is the distribution of B(u) + B(v)? The mean is 0. For the variance I get Var(B(u)+B(v)) = u+v. Is this right?

1. ∞ Ʃ (1+n)/[(n)(2n)] n=1 2. When I see that there is an n as an exponent, I think to do the ratio test. ___________________________________________________________________________ \frac{\frac{1+n+1}{(n+1)(2^{n+1})}}{\frac{1+n}{(n)(2^{n})}} = \frac{n(n+2)}{2(n+1)^{2}} =...

1. ∞ Ʃ √(n)/(n2 + 1) n=1 Find if it converges. 2. I'm wondering if I can rewrite this by bringing the n1/2 down to the denominator, making it negative... 1/(n-1/2)(n2 + 1) = 1/(n-1 + n-1/2) = n + √(n) ...And it seems to me that this one would diverge because the n value...

I'm really confused about the double sum given by my textbook. Here's what it says: If A is an nxm matrix, its length is the square root of the sum of its squares of all its entries. \left|A\right|^{2}=\sum^{n}_{i=1}\sum^{m}_{j=1}a_{i,j}^{2} The double sum is what has me caught up. How...

Question: Given a non-negative integer N, show many sets of non-negative integers (a,b,c,d) satisfy 2a+b+c+d=N Proposed (and roadblocked) solution: Case 1: 2a=0 Then there are \binom{N+2}{2} solutions (easy to prove). Case 2: 2a=2 Then there are \binom{N+2-2}{2} solutions. Case 3: 2a=4...

I'm not really sure when each of these should be done. In fact, I don't really understand the reason that we use the limit comparison test. Σ1/(n^2+1) So here I can simply say that P=2>1, so the original converges. Σ1/N^3+N^2 Here, I would say that P=3>1, implying the original...

1. Find the sum of the convergent series: ∞ Ʃ 2/[(4n-3)(4n+1)] n=12. Hm... Okay, so I started with the nth term test, and the denominator gets huge very fast. So I'm pretty sure it goes to zero. So that tells us nothing other than that it does not FOR SURE diverge. Since it has no n in the...

Homework Statement Use partial fractions to find the sum of the series.Homework Equations \displaystyle \sum^{∞}_{n=1} \frac{8}{n(n+3)} The Attempt at a Solution I end up with: \displaystyle \frac{8}{3n} - \frac{8}{3(n+3)} I am stuck here.

If $\displaystyle y=\frac{3}{4}+\frac{3*5}{4*8}+\frac{3*5*7}{4*8*12}+...\infty$. Then $y^2+2y = $

1. The problem statement Find the sum of the series: a. 1 + a cos θ + a^{2} cos 2θ + a^{3} cos 3θ + ... + a^{n} cos nθ Apparently, the answer is: \frac{a^{n+1}(a cos nθ - cos(n+1)θ) - a cos θ + 1)}{a^{2} - 2a cos θ + 1} 2. The attempt at a solution = The real part of z^{0} +...

Homework Statement Hello, My homework is online it would be to much trouble for me to type it all out it has a diagram with it. I cannot copy and paste it either. So if someone would be so kind as to message me I can send it through email as a word doc with a screen shot attached. I have...

Homework Statement Determine whether the series is convergent or divergent. If it is convergent, find its sum. \sum\limits_{n=1}^{\infty} (\frac{1}{e^n}+\frac{1}{n(n+1)}) Homework Equations The Attempt at a Solution So I found it's convergent: \sum\limits_{n=1}^{\infty}...

Hello, I am asked to evaluate the following expression using functions meshgrid, sum and dot operations in MATLAB: y = Ʃ(n=1 to N) xn*[cos(x2 + n2)/xn], where x is the vector of four equally spaced values from 1 to 2, N=10. Below is my attempt (I am quite positive it's incorrect, though): n =...

Find the limit limn→∞∑i=1 i/n^2+i^2 by expressing it as a definite integral of an appropriate function via Riemann sums ...?

Is there any general expression for the sum to nth term of the series 1/k? I know that for sufficiently large indices, a good approximation can be ln(b/a) where b and a are the upper and lower limits respectively. I've tried to do something very simple for the exact sum by constructing...

If σ(N) is the sum of all the divisors of N and τ(N) is the number of divisors of N then what is the sum of sum of all the divisors of first N natural numbers and the sum of the number of divisors of first N natural numbers? Is there any relation between σ(N) and τ(N) functions? Can I do that...

Given a large number N, do we have any formula to find the number of prime factors and their sum like τ(N) and σ(N) functions? CONDITION: One should not list the factors of N or is not allowed to factorize N since afterwards it would be just a matter of counting and addition

Homework Statement This is a two part question, though once one is solved the other should be the same process: "Write z=re^(it), where 0 < r < 1, in the summation formula and then with the aid of the theorem show that \sum r^n*cos(nt) = (r cos (t) - r^2)/(1-2r*cos(t) + r^2) when 0 < r < 1...

Hi everyone, I've been looking for the finite sum formulae of trig functions. I've found the easiest ones (sine and cosine). But the one for the tangent seems to be very hard. No mathematical tricks work. Plus I've looked it up on the internet. Nothing. I will greatly appreciate your help...

I think I'm missing something really simple here due to being out of school for a while: In a free body diagram, if I am trying to take the sum of torques about point 1, how should I deal with an applied torque at point 2? See attached sketch

Homework Statement f(x)=x^{2} from [a,b] Prove that F(x)=\frac{b^{3}}{2}-\frac{a^{3}}{2}Homework Equations The Attempt at a Solution Using the definition of an integral, we get: U(f,P)=\sum^{n}_{i=1}M_{i}(t_{i}-t_{i-1}) L(f,P)\sum^{n}_{i=1}m_{i}(t_{i}-t_{i-1}) for the function x2, how do we...

Hey, it is clear for me that \sum_{i=1}^{n} i = \frac{n(n+1)}{2} how to find a formula for \sum_{i=1}^{n} i^2 \sum_{i=1}^{n} i^3 Thanks

Homework Statement Given: https://www.physicsforums.com/attachments/56653, show that this can be written as: https://www.physicsforums.com/attachments/56651. Homework Equations Hint: https://www.physicsforums.com/attachments/56652 The Attempt at a Solution Quite confused by this...

Let W1 = {A\in MnXn(R)| A = AT} and W2 = {A\in MnXn(R)| A = -AT} Show that MnXn = W1 (+) W2 where the definition of direct sum is: V is the direct sum of W1 and W2 in symbols: V = W1 (+) W2 if: V = W1 + W2 and W1 \cap W2 = {0} Attempt: I figure I have to show each...