Sum Definition and 1000 Threads

I´m not sure, whether this little challenge has been posted before. I have searched the forum and didn´t find it. It might still be a duplicate though ... Find the sum of fractions $$\frac{2}{3\cdot5}+\frac{2\cdot4}{3\cdot5\cdot7}+\frac{2\cdot4\cdot6}{3\cdot5\cdot7\cdot9}+...$$

The problem I'd like to calculate the value of this sum: $$3 \sum^\infty_{k=1}\frac{1}{2k^2-k}$$The attempt ## 3 \sum^\infty_{k=1}\frac{1}{2k^2-k} = [k=t/2] = 3 \sum^\infty_{t=2}\frac{1}{2 \left( \frac{t}{2} \right)^2-\frac{t}{2}} = 3 \sum^\infty_{t=2}\frac{1}{ \frac{t^2}{2} - \frac{t}{2}} = 3...

If I have an asset that has a 10% chance to fail and I have ten of these assets in a basket, then what is the chance that one will fail in this basket? 10%?:partytime: What is the chance of 10 failing? 0,01%? Please also explain in some laymans terms. I am a total noob when it comes to...

So the first term of the Lagrangian is proportional to ##{F_{\mu \nu}}{F^{\mu \nu}}##. I wrote out the matrices for ##{F_{\mu \nu}}## and ##{F^{\mu \nu}}## and multiplied at the terms together and added them up, but some of the terms didn't cancel like they should have. Should I have used minus...

Hi! (Not sure which forum to pick for this question. This looked like the best one. I apologies if it is not) I have a number of functions (say m functions) with integer domain. All functions are increasing. (Increasing in the sense not decreasing, $f(n+1) \ge f(n)$.) I want an algorithm to...

Suppose I have a 1-D harmonic oscilator with angular velocity ##\omega## and eigenstates ##|j>## and let the state at ##t=0## be given by ##|\Psi(0)>##. We write ##\Psi(t) = \hat{U}(t)\Psi(0)##. Write ##\hat{U}(t)## as sum over energy eigenstates. I've previously shown that ##\hat{H} = \sum_j...

Find the exact sum of the series: $$S = \frac{1}{1\cdot 2\cdot 3\cdot 4}+\frac{1}{5 \cdot 6 \cdot 7 \cdot 8}+...$$

Given non-negative reals, $\alpha_i$, where $i = 1,2,...,n.$ Prove, that $\alpha_1+\alpha_2+...+\alpha_n \leq \frac{1}{2}$ $\Rightarrow$ $(1-\alpha_1)(1-\alpha_2)...(1-\alpha_n) \geq \frac{1}{2}.$

Homework Statement I have to determine the sum and difference of the two coefficients of drag between two individual items that make a torque AUT. I'm given the diameter of each bottle (6.35 cm), wind velocity (14.3 m/s), the forces of the lift load cells (+0.0741 N, -0.0741 N), and the force...

Hi, Let the following function: X = ∑^{L}_{k=1} f(k)/L, where f(k) is a continuous random function and L is a random discrete number. Both L and f(k) are non negative random variables. Thus, X is the average of f(k) with respect to L. Is it right to say that X equals (or approximately) to...

Prove the identity \[\sum_{j=1}^{n-1}\csc^2\left ( \frac{j\pi}{n} \right ) = \frac{n^2-1}{3 }.\]

I can't follow the solution given in my textbook to the following problem. The solution goes right off the rails on the first step. Consider a system whose Hamiltonian is given by \hat H = \alpha \left( {\left. {\left| {{\phi _1}} \right.} \right\rangle \left\langle {\left. {{\phi _2}}...

Prove that $$\sum_{n=0}^\infty \frac{n^2}{n!}=2e.$$

Prove the inequality: \[\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2} < \frac{7}{4}, \: \:\: \: n\in \mathbb{N}.\] - without using the well-known result: \[\lim_{n\rightarrow \infty }\sum_{k=1}^{n}\frac{1}{k^2} = \frac{\pi^2}{6}\]

Homework Statement "Given that ##FS f(x)=\frac{L^2}{3}+(\frac{2L}{\pi})^2⋅\sum_{n=1}^\infty \frac{(-1)^n}{n^2}cos(\frac{n\pi}{L}x)##, prove that ##\sum_{n=1}^\infty \frac{(-1)^n}{n^2}=\frac{\pi^2}{12}## and ##\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}##. Homework Equations ##FS f(x) =...

Prove $$\sum^{(N-1)/2}_{n=1}\cos\left[\frac{\pi}{N}(2n-1)\right]=\frac12$$ For $N=3,5,7...$.

Let ##M## and ##N## be connected n-manifolds, n>2. Prove that the fundamental group of ##M#N## (the connected sum of ##M## and ##N##) is isomorphic to ##\pi(M)* \pi(N)## (the free group of the fundamental groups of ##M## and ##N##) This is not for homework, I was hoping to get some insight...

Hi PF! I'm trying to compute $$S_i(\theta) = \sum_{m=odd}^{N}\frac{64}{\pi^3 m \alpha}\frac{k^2}{4k^2-m^2} \sum_{l=0}^N J_l(\alpha m) \xi_l \left( B_l(\theta)\cos(l\theta)+C_l(\theta)\sin(l\theta) \right)$$ where the ##i## subscript appears since $$B_l = \int_0^{2\pi} \psi_i(\theta)...

T 4–4 Deposits needed to accumulate a future sum Judi wishes to accumulate \$8,000 by the end of 5 years by making equal annual end-of-year deposits over the next 5 years. If Judi can earn 7% on her investments, how much must she deposit at the end of each year to meet this goal...

Homework Statement Given X_1,\dots,X_n a simple random sample with normal variables (\mu, \sigma^2). We assume \mu is known but \sigma^2 is unknown. The hypothesis is \begin{cases} H_0: & \mu=\mu_0 \\ H_1: & \mu=\mu_1 > \mu_0 \end{cases} Determine the rejection region R...

Evaluate the sum:$$\sum_{n=0}^{2017}\frac{1}{3^n+\sqrt{3^{2017}}}$$

Let $S_n$ be the sum of lengths of all the sides and all the diagonals of a regular $n$-gon inscribed in a unit circle. (a). Find $S_n$. (b). Find $$\lim_{{n}\to{\infty}}\frac{S_n}{n^2}$$

Homework Statement http://i66.tinypic.com/aesd1u.png can someone explain to me how can i get the limit using riemann sum especially the starred part? i was so confused thanks! Homework Equations The Attempt at a Solution attempt at a solution in the picture

In the picture above, there are three balls in separated small elevators. Elevator A lifts the ball upwards, elevator B stays still, elevator C moves the ball downwards, all in constant speed. (And this is a model, a simplification of the reality, we assume no other forces on the balls other...

Homework Statement Suppose I have a system which contains two bodies m1 and m2 with initial velocities v1 and v2 , respecitvely. they hurl toward each other and make an inelastic collision. such that they are now one body of mass m1 + m2 I know that the difference in momentum is...

∑ab is needed but is impractical to implement. Specifically ∑i ai.10i-|i| in any form where I can work with ∑i ai = α and ∑i 10i-|i| separately. Is there a homomorphic function I can run it through such that ∑ab can be expressed as ∑a∑b? Note: for current problem i cannot simply set it up...

Homework Statement i want to find limit value using riemann sum \lim_{n\to\infty}\sum_{i = 1}^{2n} f(a+\frac{(b-a)k}{n})\cdot\frac{(b-a)}{n}= \int_a^b f(x)dx<br> question : <br> \lim_{h \to \infty} =\frac{1}{2n+1}+\frac{1}{2n+3}+...+\frac{1}{2n+(2n-1)}<br> Homework EquationsThe Attempt at a...

Homework Statement Homework EquationsThe Attempt at a Solution The probability that ##X_1 ## is between ## X_1 ## and ## X_1 + dX_1 ## and ##X_2 ## is between ## X_2 ## and ## X_2 + dX_2 ## and so on till the nth variable is dP(##X_1, X_2, ..., X_n) = p ( X_1) p( x_2) p(X_3)...p(X_n) dX_1...

I apologize for the simplicity of the question (NOT homework). This is a statistical question (not necessarily a quantum mechanical one). If I have an initial probability function with an associated expected value and then a second probability function is superimposed on the initial...

Okay, so I need to prove this. I thought I would be using induction, right? First we can consider the base case, which is simple. Next we have to do the induction step. I think we consider one case where each a=1. Then we have 1<=1. Then consider that they are not all 1. I can't remember...

Hello all, I'm a bit baffled by the fact that the various quite different components of the SM Lagrangian (or other systems, btw) are simply summed up, without even one ponderation coefficient, in the total Lagrangian. I know one reason it is like that is that... it works in practice, but I...

I'm starting with my self studying of math with Algebra I. The text I'm using is Gelfand and Shen's Algebra. I'm at the point where it talks about the Formula for the Square of the Sum, The Square of the Distance Formula, and The Difference of Squares Formula. In going over this, I understand...

$\tiny{hcc8.12}$ $\textsf{Find the sum of vectors $(10, \, 45^o)$ and $(7, \, 150^o)$}\\$$\begin{align*}\displaystyle \textsf{magnitude} &=\sqrt{(10\cos45^o - 7\cos150^o)^2 + (10\sin45^o - 7\sin150^o)^2} \approx 12.114 \end{align*}$ ok an online vector sum calculator returned 10.619 so...

In my quest to understand the Euler-Lagrange equation, I've realized I have to understand the chain rule first. So, here's the issue: We have g(\epsilon) = f(t) + \epsilon h(t). We have to compute \frac{\partial F(g(\epsilon))}{\partial \epsilon}. This is supposed to be equal to \frac{\partial...

I am trying without success to provide a rigorous proof for the following exercise: Show that the sum of a rational number and an irrational number is irrational.Can someone please help me with a rigorous solution ...I am working from the following books: Ethan D. Bloch: The Real Numbers and...

Homework Statement I am trying without success to provide a rigorous proof for the following exercise: Show that the sum of a rational number and an irrational number is irrational. Homework Equations I am working from the following books: Ethan D. Bloch: The Real Numbers and Real Analysis...

Hi PF! I'm trying to put the first derivative of the modified Bessel function of the first kind evaluated at some point say ##\alpha## in a sum where the ##ith## function is part of the index. What I have so far is n=3; alpha = 2; DBesselI[L_, x_] := D[BesselI[L, x], {x, 1}] Sum[BesselI[L...

$\tiny{242 .10.09.8}\\$ $\textsf{Express the integrand as a sum of partial fractions and evaluate integral}$ \begin{align*}\displaystyle I&=\int f \, dx = \int\frac{\sqrt{16+5x}}{x} \, dx \end{align*} \begin{align*}\displaystyle f&=\frac{\sqrt{16+5x}}{x}...

Homework Statement Trying to find the sum of (-1)3n+1/(2n-1)3. by using term-by-term integration on the cosine Fourier series x= L/2-4L/π2∑cos(((2n-1)πx)/L)/(2n-1)2. Homework Equations Shown below The Attempt at a Solution When integrating and substituting Lx/2 for x's sine Fourier series I...

Evaluate the sum:\[\sum_{n = 0}^{\infty}\frac{n}{n^4+n^2+1}\]

√(2-√(2^(2)-1))+√(4-√(4^(2)-1))+√(6-√(6^(2)-1))+...+√(80-√(80^(2)-1)) How the find it's value

Hi I was wondering if someone could tell me the equation for finding out what the windage and friction losses are based on the information below. I've drawn the graph and I can work out the I/0, R0 and X0A four-pole, star-connected, squirrel-cage induction motor operates froma variable voltage...

Homework Statement Verify that the sum of three quantities x, y, z, whose product is a constant k, is maximum when these three quantities are equal. Homework Equations w = x + y + z k = x * y * z The Attempt at a Solution Assuming that my understanding of the question is correct i.e. that we...

Homework Statement Given that a1+a2+a3+... +a7=1 and that all aj is smaller than 1 and greater than 0 and variable , find the maximum value of 6a1+5a2+4a3+...+ a6 Homework Equations no further constraints The Attempt at a Solution I have no idea how to solve this problem specifically, but I...

Hello everybody, How can i check if this sum is converges?

I decided to review a little trigonometry. Why does tan(x + pi/2) = -cotx? I cannot use the tangent of a sum formula because tan(pi/2) does not exist. How about tan(x + pi/2) = [sin(x + pi/2)]/[cos(x + pi/2)] and then apply the addition rules for sine and cosine?

Hello all. Thanks for taking up your time to read my post. I checked the currents that passed through several LEDs for MIC2287CBD5 as the datasheet of http://www.kynix.com/uploadfiles/pdf2286/MIC2287CBD5.pdf at a particular voltage. They varied from 12.0 to 15.1 mA at 3.0 V. I then connected...

Homework Statement evaluate ## \sum _{n=1}^{\infty }4^{\frac{1}{n}}-4^{\frac{1}{n+2}}## . https://holland.pk/uptow/i4/fc981b864d95a636c4f08b9deb209cd6.png Homework Equations telescoping series: sum = infinite lim (a1-a(n+1)) S=a/(1-r) The Attempt at a Solution as the latter function is of...

Homework Statement Suppose V=U⊕W. Prove that V0=U0⊕W0. (V0= annihilator of V). Homework Equations (U+W)0=U0∩W0 The Attempt at a Solution Well, I don't see how this is possible. If V0=U0⊕W0, then U0∩W0={0}, and since (U+W)0=U0∩W0, it means (U+W)0={0}, but V=U⊕W, so V0={0}. I don't think this...

Homework Statement determine the number of pairs of integers (a,b) 1≤b<a<200 such that the sum ## (a+b) + ( a-b) + ab + \frac a b\ ## is the square of an integer i have the solution to the problem this was the given solution the given equation is equivalent to ## \frac {a*(b+1)^2} b\\ ##...