Sum Definition and 1000 Threads

$a,\,b$ and $c$ are positive real numbers such that $25bc+9ac+ab=9abc$ and $a+b+c=9$. Find the sum of all possible values of $abc$.

Homework Statement I have this exercise: Calculate ##\sum\limits_{k=0}^\infty t^{k}sin{(kx)}## Where x and t are real and t is between 0 and 1. Homework Equations ? The Attempt at a Solution The ratio test says that this sum does have a limit, and tk obviously converges, as t is between 0 and...

Hi I have 2 linear integral operators $(Ku)(x)=\int_{a}^{b} \,k(x,t)u(t)dt$ $(Mu)(x)=\int_{a}^{b} \,m(x,t)u(t)dt$ I am defining $||K||=max([x\in[a,b]\int_{a}^{b} \,|k(x,t| dt$ same for $L$ when does $||K+M||=||K||+||M||$ thanks sarrah

Find the minimum of $\large \log_{a_1}\left(a_2-\dfrac{1}{4}\right)+\log_{a_2}\left(a_3-\dfrac{1}{4}\right)+\cdots+\log_{a_n}\left(a_1-\dfrac{1}{4}\right)$ where $a_1,\,a_2,\cdots,a_n$ are real numbers in the interval $\left(\dfrac{1}{4},\,1\right)$.

Let $x \in R - \{0\},$ where $R$ is a domain. Define $T_x(M) = \{m \in M \ | \ x^n m=0 \ \ \mathrm{for \ some} \ n \in \mathbb{N}\}$ as the $x$-torsion of $M.$ I know that $T_x(M \oplus N) = T_x(M) \oplus T_x(N)$ for $R$-modules $M,N$ only if $R$ is a PID. But I can't think of a...

Letting $X$ be a ring and $K$ be an $X$-module, I need to show that **if** $K \cong A \times B$ for some $X$-modules $A,B$, **then** $\exists$ submodules $M'$ and $N'$ of $K$ such that: $K=M' \oplus N'$ $M' \cong A$ $N' \cong B.$----------I understand the concepts of internal and external...

I'm not sure if this is the right place to post this in, but I'm trying to recreate the "Deformation of water by a magnetic field" experiment by Chen et al. The PDF version of the paper can be accessed via Google (for some reason it won't let me provide a direct link). On the 2nd page of the...

Homework Statement You have series expansions of the function f(x) = 0 from 0 to .5, and 1 from .5 to 1 : the halfrange cosine series, the half-range sine series, and the Fourier series. For each of these series, find the actual sum of the series at x = 0, and x =1/2, and x =1 Homework...

Hi everyone, I am trying to solve a problem (related to pharmacokinetics) that requires finding t for an equation like the following: A e^{-a t} - B e ^{-b t} + C e^{-c t} = 0 where A, a, B, b, C and c are all real positive numbers (they are constants related to the absorption, uptake, efflux...

It can be shown that the following sum: $$S=\sum_{k=1}^{89}\left(\sin^6\left(k^{\circ}\right)\right)$$ is rational. Find the value of $S$. (Callme)

Find a closed form expression for $$\sum_{k=1}^{n^2}\dfrac{n-\lfloor\sqrt{k-1}\rfloor}{\sqrt{k}+\sqrt{k+1}}$$.

Homework Statement We have an infinite plane of width 2b made of a magnetic, conducting material (μr >> 1, σ >> 1). Two monochromatic electromagnetic plane waves, with magnetic excitation vector amplitude Hs approach it, each one traveling towards one of its two faces. Find the current density...

Let $p$ be an odd prime. Prove that $p$ is the sum of 2 consecutive squares i.e. $p=a^2+(a+1)^2$ if and only if $p$ has the form $p=\dfrac{u^2+1}{2}$.

Hi everybody, I'm facing a tricky summation problem. The problem is (both mathematically and physically) related to this thread I started a while ago. I'm starting a new thread because the functions are not exactly the same, and I have (perhaps) made some progress, using Euler-Maclaurin sum...

Is it possible to write $5^{64}-3^{64}$ as the sum of two squares?

Hello all, I have been meditating on this for a while, but can't seem to understand how this simplification came to be. Any help will be greatly appreciated. So, here is what we start with: ##\mathop{\sum_{k=0}^m\sum_{l=0}^n}_{m{\geq}n} x(k,l)## We also know that: l (lower case L) = n-m+k and...

Evaluate the infinite series $1-\dfrac{2^3}{1!}+\dfrac{3^3}{2!}-\dfrac{4^3}{3!}+\cdots$.

I know that the reduced mass, μ, of an object is: \mu = \frac{1}{\frac{1}{m_1} + \frac{1}{m_2}} \mu = \frac{m_1 m_2}{ m_1 + m_2 } But is there a general formula (or a simplified expression) for finding the value of: \frac{1}{ \frac{1}{x_1} + \frac{1}{x_2} + ... + \frac{1}{x_n} } ? Thank you.

The question provides a table of values for time and velocity. Part c of the question asks to use a Riemann sum to approximate (not specifying which one). Part d asks what the answer to part c represents and to explain my reasoning. The answer that I got for the sum is 58.5 feet, but I do not...

I think I understand the basic ideas of the sum and difference formulas, I just don't get how to break down say, pi/7 into a form that could be worked with. I could convert it to a degree then back again once I have my answer, but that seems like a lot more work than is necessary. If it were 100...

$\sum_{i=1}^3 2 i = 12$

Homework Statement The question arises from this quote from wikipedia's article about kronecker product: Kronecker sums appear naturally in physics when considering ensembles of non-interacting systems. Let Hi be the Hamiltonian of the i-th such system. Then the total Hamiltonian of the...

Hey! :o We suppose that a force $\overrightarrow{F}$ (for example, the gravity) is applied vertically downwards to an object that is placed at a plane which has an angle of $45^{\circ}$ with the horizantal direction. Express this force as a sum of a force that acts parallel to the plan and of...

Homework Statement Given an integer n and an angle θ let Sn(θ) = ∑(eikθ) from k=-n to k=n And show that this sum = sinα / sinβ Homework Equations Sum from 0 to n of xk is (xk+1-1)/(x-1) The Attempt at a Solution The series can be rewritten by taking out a factor of e-iθ as e-iθ∑(eiθ)k from...

hi guys. i have 2 questions, how do solve this problem with formula [shortcut] : please, see attachment file.. thanks for your helping.. susanto3311

hello all... would you help me out, how to sum 3 questions problem.. please, see my picture attachment.. thanks for your helping... susanto3311

Hello everyone! My question is: Where is sum of divergent series and divergent integrals used in physics? What it all means? Where can I find examples of divergent integrals? Is there a book of problems for physicists? I am mathematician. I developed a method for summing divergent series...

Homework Statement The ratio of sums of 2 AP for n terms each is ## \frac{3n + 8}{7n + 15}## that is $$ {\frac{s_a}{s_b}} = \frac{3n + 8}{7n + 15} $$ find the ratio of their 12th terms. $$ Required= \frac{a₁_a+(n-1)d_a}{a_b + (n-1)d_b}$$Homework Equations Tn = a + (n-1)dThe Attempt at a...

I have a = {a1, a2, .., a1000}, where this set forms a distribution of photoelectrons (pe) seen by a particular photomultiplier tube (pmt) over 1000 repeated events. I then have N sets of these (N pmts), each containing 1000 pe values which I believe are indeed random and independent. So a, b...

Homework Statement When is |x+y|=|x|+|y| for arbitrary non-zero vectors x,y∈Rn ie, when does equality hold for the well known inequality |x+y|≤|x|+|y| Homework Equations |x|2=<x,x>=Σixi2 |x+y|≤|x|+|y| The Attempt at a Solution squaring both sides of the inequality we have...

In a differential protection relay there is a setting of a threshold for the sum of amps (or the squares). In what way would that be of interest? The relay is an Areva P521. Also, the relay can display the ratio between the positive sequence current and the negative. What information does that...

Question: Convert to Sum of Products (p+q’+r+s’). (p’+q+r+s’). (p’+q’+r+s’) = ((p+q’+r+s’)’ + (p’+q+r+s’)’ + (p’+q’+r+s’)’)’ (Demorgans each Sum-Term) = ((p’.q.r’.s) + (p.q’.r’.s) + (p.q.r’.s))’ (Factor p.r’.s) = ((p’.q.r’.s) + (p.r’.s).(q’ + q))’ (Inverse Law) = ((p’.q.r’.s) + (p.r’.s))’...

Hi PF! I'm reading my math text and am looking at the heat eq ##u_t = u_{xx}##, where we are are given non-homogenous boundary conditions. We are solving using the method of eigenfunction expansion. Evidently we begin by finding the eigenfunction ##\phi (x)## related to the homogenous...

Hi all, given the sum of a series and a single term how would one find the nth term? Any help appreciated.

Homework Statement inputs x1(t) = cos(ω1t), x2(t) = cos(ω2t). Show that output g(t) (sum of x1 + x2) = 0.5cos[(ω2-ω1)t] + 0.5cos[(ω2+ω1)t] Homework Equations included in upload of attempted solution. Trig identities. The Attempt at a Solution Uploaded in pdf. A lot more has been done on the...

Homework Statement Find the sum of the following series: Σ n*(1/2)^n (from n = 1 to n = inf). Homework Equations I know that Σ r^n (from n = 0 to n = inf) = 1 / (1 - r) if |r| < 1. The Attempt at a Solution [/B] I began by rescaling the sum, i.e. Σ (n+1)*(1/2)^(n+1) (from n = 0 to n =...

I'm trying to investigate this statement: The sum of n consecutive numbers is always divisible by n. I've found already that it's only true when the total amount of numbers is an odd number. I've also found that the median and mean are the same with consecutive numbers. I can not prove that the...

Hello, I've read that the force F we use in the pressure formula P=F/A is an average of all microscopic forces exerted by the gas on the wall of the container over a period of time. Is it true? I thought it was a SUM rather than an average? I assume that in that definition the microscopic force...

Homework Statement Homework Equations Down The Attempt at a Solution As you see in the solution, I am confused as to why the sum of residues is required. My question is the sum: $$(4)\cdot\sum_{n=1}^{\infty} \frac{\coth(\pi n)}{n^3}$$ Question #1: -Why is the beginning n=1 the residue...

Homework Statement The value of ##\sqrt{12+\sqrt{12+\sqrt{12+ ...}}}## is Homework EquationsThe Attempt at a Solution No Clue

I am having a hard time grasping contact curvature sums. Can someone give me a link to where there is a guide or a youtube video? or can someone help me here please. Here are the equations: 1/Rx = 1/rax + 1/rbx 1/Ry = 1/ray + 1/rby 1/R = 1/Rx + 1/Ry The question is this: The ball-outer...

Sorry for the confusing tittle but I could not explain it better. Here is what I am trying to ask: When you have 2 axis(x and y) such as the image below, the sum of the two angles, a and b will always be equal to 90 degrees. a + b = 90degrees However when you add a 3rd axis(x, y and z, making...

Under what conditions are the symbols \bigoplus and \times intechangangable?

Homework Statement Find the sum of the series from k=0 to infinity of ((4^k)-(3^k))/(5^k) Homework Equations I'm not sure exactly. I know the test for divergence is if lim n approaches infinity of the function from m=1 to infinity does not equal 0 then the series cannot diverge The Attempt...

Homework Statement Let ## S_k , k = 1,2,3,…,100 ## denote the sum of the infinite geometric series whose first term is ## \frac{k-1}{k!} ## and the common ratio is ##\frac {1}{k}##. Then value of ##\frac {100^2}{100!} + \sum\limits_{k=1}^{100} | (k^2 - 3k + 1)S_k | ## is Homework Equations...

Dear Friends! Is sum of deviations from median always minimum,in comparison to deviations from mean,mode or any other observation?Why?

It's an amazing question to find sum of last two digits of 27^27 by not using that wolframalpha or calculators.I think binomial theorem would be of help here but not able to apply.can anyone tell me the answer by any method?

Prove that for every positive integer $a$, the integer $5^{4a}+5^{3a}+5^{2a}+5^a+1$ is composite.

I saw that somewhere and it is supposed to be something Fourier came up with but I can't find somewhere why... Please explain (with mathematical description if possible)

Homework Statement If Y=X1+X2+...+XN prove that <Y>=<X1>+<X2>+...+<XN> Homework Equations <Y>=∫YP(Y)dY over all Y. The Attempt at a Solution I only seem to be able to show this if the Xi are independent, and I also think my proof may be very wrong. I basically have said that we can write the...