Sum Definition and 1000 Threads

I have encountered the name "sum rules" in many places, such as EM, acoustics,... But I understand none. In what kind of situation will a sum rule occurs? What does it said exactly?

Consider the differential equation $$y''+ay'+by=0$$ We have analytical solutions for this equation. There are three cases to consider based on the discriminant of the characteristic polynomial associated with the equation. $$\Delta=a^2-4b$$ I just want to discuss the case where $$\Delta...

Just looking at the summand, I can see that the function is ln(pi/4 + x^2) as the (i pi/2n) term is the 'x' term. How do I determine the limits of the integral, however? I was thinking about using the lower bound of the summation --> this given the (pi / 2n)^2 term, implying that nothing was...

Hello all, I'm in the process of learning about 3 phase power and how to wire loads to a generator. I've searched high and low with as many sentence structures as I could think of in the google search bar and I can't seem to find the answers I'm looking for. I'm really hoping you guys can...

Find, with proof, the sum of the alternating series $$\sum_{n = 0}^\infty \frac{(-1)^n}{(2n+1)^3}$$

My interest is on the (highlighted part in yellow ) of finding the partial fractions- Phew took me time to figure out this out :cool: My approach on the highlighted part; i let ##(kr+1) =x ## then, ##\dfrac{1}{(kr+1)(kr-k+1)} = \dfrac{1}{x(x-k)}## then...

There is a mistake on this textbook (The mistake is pretty obvious) but hey I hope I did not miss something... ought to be ##40425 - 19600##

##(1+x)^n=1+C_1x+C_2x^2+C_3x^3...+C_nx^n## Let ##x=1##, hence ##2^n=1+C_1+C_2+C_3...+C_n## which is equal to the sum of the coefficients. I originally thought the sum of the coefficients would be ##2^n-1## since the very first term ##1## is just a number and has no variable. But apparently...

Hi! I am having trouble finalizing this problem. The interval is given so we know that a = 1 and b = 2. From there you can figure out that ∆x = 1/n, xiR = 1 + i/n. Using logarithmic properties, I rearranged the expression and wrote (1 + i/n)(1/n)ln[(n + i)/n]. I can guess that the function is...

Hi! I understand that this is an expanded Riemann sum but I'm having trouble determining its original form. I don't actually have any ideas as to how to find it, but I know that once I determine the original form of the Riemann sum, I will be able to figure out the values for a, b, and f. If...

Problem Statement : The statement appeared on a website where a different problem was being solved. I got stuck at the (first) statement in the solution that I posted above 👆. Here I copy and paste that statement from the website, which I cannot show : Attempt : To save time typing, I write...

Lemma Let f be a bounded function on [a,b]. If P & Q are partitions of [a,b] and P ##\subseteq## Q , then L(f,P) ##\leq## L(f,Q) ##\leq## U(f,Q) ##\leq## U(f,P) . Question is "How can P have bigger upper darboux sum than Q while it is a subset of Q"

I want to know if there is any proper relation between the angles of a vector with the three dimensional coordinate axes, if the angles are ,α , β and γ, will the sum of α, β and γ be 180 degress that is α + β + γ = 180°,m finding the same to be true in a 2 D case where α + β = 90° and γ =...

Hi, I'm struggling with understanding the idea of tensor product and direct sum beyond the very basics. I know that direct sum of 2 vectors basically stacks one on top of another - I don't understand more than this . For tensor product I know that for a product of 2 matrices A and B the tensor...

Attempt; ##\dfrac{1}{r(r+1)(r+2)} -\dfrac{1}{(r+1)(r+2)(r+3)}=\dfrac{(r+3)-1(r)}{r(r+1)(r+2)(r+3)}=\dfrac{3}{r(r+1)(r+2)(r+3)}## Let ##f(r)=\dfrac{1}{r(r+1)(r+2)}## ##f(r+1)= \dfrac{1}{(r+1)(r+2)(r+3)}## Therefore ##\dfrac{3}{r(r+1)(r+2)(r+3)}## is of the form ##f(r)-f(r+1)## When...

My attempt; ##r^2+r-r^2+r=2r## Let ##f(r)=(r-1)r## then it follows that ##f(r+1)=r(r+1)## so that ##2r## is of the form ##f(r+1)-f(r)##. When ##r=1;## ##[2×1]=2-0## ##r=2;## ##[2×2]=6-2## ##r=3;## ##[2×3]=12-6## ##r=4;## ##[2×4]=20-12## ... ##r=n-1##, We shall have...

From my physical problem, I ended up having a sum that looks like the following. S_N(\omega) = \sum_{q = 1}^{N-1} \left(1 - \frac{q}{N}\right) \exp{\left(-\frac{q^2\sigma^2}{2}\right)} \cos{\left(\left(\mu - \omega\right)q\right)} I want to know what is the sum when N \to \infty. Here...

I've came across the two following theorems in my studies of Probability Generating Functions: Theorem 1: Suppose ##X_1, ... , X_n## are independent random variables, and let ##Y = X_1 + ... + X_n##. Then, ##G_Y(s) = \prod_{i=1}^n G_{X_i}(s)## Theorem 2: Let ##X_1, X_2, ...## be a sequence of...

Hi ! In a Feynman diagram, can we consider that the propagator specifying the transition amplitude of a particle (let's say, of a "real" electron, or of a "virtual" photon) between two points or two vertices, is in fact itself the sum of a multiplicity of probability amplitudes, each one...

Since we are adding numbers produced according to a fixed pattern, there must also be a pattern (or formula) for finding the sum. Hi, We use this method to find the ##S_n##. I don't understand how the sum will also be in a pattern. Can someone please explain this line in bold?

I am refreshing on this; of course i may need your insight where necessary...I intend to attempt the highlighted...this is a relatively new area to me... For part (a), We shall let ##f(x)=\dfrac{1}{x(2-x)}##, let ##g(x)## be the even function and ##h(x)## be the odd function. It follows...

Find all triples (a, b, c) of positive integers such that ##a^3+b^3+c^3=(abc)^2##.

See the work below: I feel like it that I did it correctly. I feel like I skip a step in my induction. Please point any errors.

If a and b are two real numbers such that ##a^3-3a^2+5a=1## and ##b^3-3b^2+5b=5##, evaluate a + b.

im thinking i should just integrate (binominal distribution 1-2000 * prime probability function) and divide by integral of bin. distr. 1-2000. note that I am looking for a novel proof, not just some brute force calculation. (this isn't homework, I am just curious.)

Hello. As is known, we can neglect high-order term in expression ##f(x+dx)-f(x)##. For ##y=x^2##: ##dy=2xdx+dx^2##, ##dy=2xdx##. I read that infinitesimals have property: ##dx+dx^2=dx## I tried to neglect high-order terms in integral sum (##dx^2## and ##4dx^2## and so on) and I obtained wrong...

Hello, I would like to know, if there's a closed form solution to the following problem: Given a sum of say, 3 sines, with the form y = sin(a.2.PI.t) + sin(b.2.PI.t) + sin(c.2.PI.t) where a,b,c are constants and PI = 3.141592654 and the periods in the expression are multiplication signs, what...

How do I merge ##x^n + \displaystyle\sum^x_{k=1} \frac{d^k}{d^kx} \frac{x^ny^k}{k}##? I tried changing the denominator of the summand to ##k+1## and make ##k## go from zero, but I had to divide by zero when k equaled one.

I got something like this, but I'm not sure it is correct, because if it has the same order of convergence as trapezoidal rule which is 2, it should yield the same result as trapezoidal rule but mine doesn't (?). For example sin(x) for [0,1], n with trapezoidal rule = 0.420735... With my own...

Find question and solution here Part (i) is clear to me as they made use of, $$\sum_{r=n+1}^{2n} u_r=\sum_{r=1}^{2n} u_r-\sum_{r=1}^{n} u_r$$ to later give us the required working to solution... ... ##4n^2(4n+3)-n^2(2n+3)=16n^3+12n^2-2n^3-3n^2=14n^3+9n^2## as indicated. My question is on...

I am refreshing on this...Have to read broadly...i will start with (b) then i may be interested in alternative approach or any correction that may arise from my working. Cheers. Kindly note that i do not have the solutions to the following questions... For (b), we know that, say, if ##x=α##...

Let ## n ## be an integer. Now we consider two cases. Case #1: Suppose ## n ## is even. Then ## n=2k ## for some ## k\in\mathbb{N} ##. Thus ## n^{5}=(2k)^{5}=32k^{5}\equiv 0 \pmod 4 ##. Case #2: Suppose ## n ## is odd. Then ## n=4k+1 ## or ## n=4k+3 ## for some ## k\in\mathbb{N} ##. Thus ##...

We have a matrix ##M = \ket{\psi^{\perp}}\bra{\psi^{\perp}} + \ket{\varphi^{\perp}}\bra{\varphi^{\perp}}## The claim is that the eigenvalues of such a matrix are ##\lambda_{\pm}= 1\pm |\bra{\psi}\ket{\varphi}|## Can someone proof this claim? I have been told it is self-evident but I've been...

Proof: Let ## a>5 ## be an integer. Now we consider two cases. Case #1: Suppose ## a ## is even. Then ## a=2n ## for ## n\geq 3 ##. Note that ## a-2=2n-2=2(n-1) ##, so ## a-2 ## is even. Applying Goldbach's conjecture produces: ## 2n-2=p_{1}+p_{2} ## as a sum of two primes ## p_{1} ## and ##...

I do not understand why electric fields cannot cross. Can't you just sum the two electric fields vectors to get a net electric field?

Proof: Suppose ## p ## and ## p+2 ## are twin primes such that ## p>3 ##. Let ## p=2k+1 ## for some ## k\in\mathbb{N} ##. Then we have ## p+(p+2)=2k+1+(2k+1+2)=4k+4=4(k+1)=4m ##, where ## m ## is an integer. Thus, the sum of twin primes ## p ## and ## p+2 ## is divisible by ## 4 ##. Since ##...

Hi there. Let's say I have the following relationship: x = a + b*z + c*y z is distributed normally y is distributed according to a different distribution, say exponential Is there a way to figure out what is the distribution of x? Thanks!

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if I select two integers at random between 1 and 1,000, what is the probability that their sum will be prime?

Using the given rule for the ##x_n##, write $$ \sum_l y_l = x_1 + \frac{1}{2} x_2 + \frac{1}{6} x_2 + \frac{1}{3} x_3 + \frac{1}{12} x_2 + \frac{1}{12} x_3 + \frac{1}{20} x_2 + \cdots + \frac{1}{n} x_n $$ $$ = x_1 + \sum_{n=2}^\infty \frac{1}{n(n-1)} x_2 + \sum_{n=3}^\infty \frac{2}{n(n-1)} x_3...

Summary:: summation of the components of a complex vector Hi, In my textbook I have ##\widetilde{\vec{E_t}} = (\widetilde{\vec{E_i}} \cdot \hat{e_p}) \hat{e_p}## ##\widetilde{\vec{E_t}} = \sum_j( (\widetilde{\vec{E_{ij}}} \cdot {e_{p_j}}*) \hat{e_p}## For ##\hat{e_p} = \hat{x}##...

Proof: Suppose n is an integer such that ## n>11 ##. Then n is either even or odd. Now we consider these two cases separately. Case #1: Let n be an even integer. Then we have ## n=2k ## for some ## k\in\mathbb{Z} ##. Consider the integer ## n-6 ##. Note...

I can't understand how this approximation works ##\sum_{k=0}^m\left(\frac{k}{m}\right)^n\approx\int_0^m\left(\frac{x}{m}\right)^ndx\tag{1}##Can you please help me

Hi, I'm wondering if I have an expression for the scalar magnetic potential (V_in) and (V_out) inside and outside a magnetic cylinder and the potential is continue everywhere, which mean ##V^1 - V^2 = 0## at the boundary. Does it means that ##V^1 - V^2 = V_{in} - V_{out} = 0## ?

For part (i), ##(x-α)(x-β)=x^2-(α+β)x+αβ## ##α+β = p## and ##αβ=-c## therefore,##α^3+β^3=(α+β)^3-3αβ(α+β)## =##p^3+3cp## =##p(p^2+3c)## For part (ii), We know that; ##tan^{-1} x+tan^{-1} y##=##tan^{-1}\left[\dfrac...

For part a, We have ##α+β=b## and ##αβ =c##. It follows that, ##(α^2 + 1)(β^2+1)=α^2β^2+α^2+β^2+1)## =##α^2β^2+(α+β)^2-2αβ +1## =##c^2+b^2-2c+1## =##c^2-2c+1+b^2##...

Given are non-negative integer variables ##x##, ##y## and ##z##. I am trying to deduce the absolute difference between a certain value of ##C=x^2+y^2+z^2## and the very next smallest increase in ##C## possible. I'd like to do this so I can (dis)prove the following: Whether small absolute...

Problem statement : I copy and paste the statement of the problem from the text. (Given ##\boldsymbol{a+b+c=0}##) Attempt : I am afraid I couldn't make any meaningful progress. With ##a = -(b+c)##, I substituted for ##a## in the whole of the L.H.S, both numerators and denominators. I multiplied...