What is Sum: Definition and 1000 Discussions

Sum, sumu, sumon, and somon (Plural: sumd) are the lowest level of administrative division used in China, Mongolia, and Russia. The word sumu is a direct translation of a Manchu word niru, meaning ‘arrow’ Countries such as China and Mongolia, have employed the sumu administrative processes in order to fulfil their nations economic, social and political goals. This system was acted in the 1980s after the Chinese Communist Party gained power in conjunction with their growing internal and external problems. The decentralisation of government included restructuring of organisational methods, reduction of roles in rural government and creation of sumu’s.

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  1. Rectifier

    What Is the Value of the Infinite Sum \(3 \sum_{k=1}^\infty \frac{1}{2k^2-k}\)?

    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...
  2. Observeraren

    B What is the sum of multiple probabilities

    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...
  3. Gene Naden

    I Sum Maxwell Lagrangian 1st Term: Use Minus Signs?

    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...
  4. Pereskia

    MHB Algorithm: maximize sum of increasing functions

    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...
  5. M

    QM: Writing time evolution as sum over energy eigenstates

    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...
  6. lfdahl

    MHB Find the exact sum of the series 1/(1⋅2⋅3⋅4)+1/(5⋅6⋅7⋅8)+....

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

    MHB Can Sum to Product Inequalities Hold for Non-Negative Reals?

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

    Finding the sum and difference of two drag coefficients

    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...
  9. N

    A Averaging over the upper sum limit of a discrete function

    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...
  10. lfdahl

    MHB What is the Proof for the Trigonometric Sum Identity?

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

    I Squaring a Sum of Ket-Bra Operators

    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}}...
  12. lfdahl

    MHB Prove the sum identity ∑n2n=2e.

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

    MHB Proving Inequality: \(\frac{1}{n^2}\) Sum < \(\frac{7}{4}\)

    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: \[\lim_{n\rightarrow \infty }\sum_{k=1}^{n}\frac{1}{k^2} = \frac{\pi^2}{6}\]
  14. Eclair_de_XII

    How to convert an alternating sum into a positive sum?

    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) =...
  15. Greg

    MHB Trigonometric Sum Prove: N=3,5,7...

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

    A Connected sum of manifolds and free group isomorphisms

    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...
  17. M

    MATLAB Is My Code Correct for Computing This Double Sum?

    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)...
  18. karush

    MHB T 4–4 Deposits needed to accumulate a future sum

    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...
  19. G

    Minimize the sum of Type I and Type II errors

    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...
  20. lfdahl

    MHB Find The Sum ∑(1/[3^n+√(3^(2017)]

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

    MHB What is the Sum of Lengths for a Regular n-gon Inscribed in a Unit Circle?

    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 $$\lim_{{n}\to{\infty}}\frac{S_n}{n^2}$$
  22. D

    Calculating the Limit Using Riemann Sum with Starred Part?

    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
  23. K

    B Relation between sum of the forces and energy

    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...
  24. C

    Inelastic collision and the sum of internal forces

    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...
  25. N

    I Homomorphism of an elementwise sum and dot product

    ∑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...
  26. D

    Find the limit using Riemann sum

    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...
  27. Pushoam

    Avg. of sum of independent variables

    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...
  28. redtree

    B The expectation value of superimposed probability functions

    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...
  29. J

    MHB Proving the Sum of a Sequence Equals n Using Induction

    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...
  30. J

    I Uncovering the Mystery Behind SM Lagrangian Sums

    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...
  31. D

    B The Square of the Sum Formula and real world scenarios

    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...
  32. karush

    MHB Hcc8.12 Find the sum of vectors

    $\tiny{hcc8.12}$ $\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...
  33. Avatrin

    I Rigorously understanding chain rule for sum of functions

    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...
  34. Math Amateur

    MHB Sum of a rational number and an irrational number ....

    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...
  35. Math Amateur

    Sum of a rational number and an irrational number ....

    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...
  36. M

    Mathematica Bessel function derivative in sum

    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...
  37. karush

    MHB 242 .10.09.8 Express the integrand as a sum of partial fractions and evaluate integral

    $\tiny{242 .10.09.8}\\$ $\textsf{Express the integrand as a sum of partial fractions and evaluate integral}$ \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}...
  38. John Jacke

    Finding the sum of an infinite series using Fourier

    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...
  39. lfdahl

    MHB Evaluate the sum ∑n/[n^4+n^2+1]

    Evaluate the sum:\[\sum_{n = 0}^{\infty}\frac{n}{n^4+n^2+1}\]
  40. Mathysics29

    Olympiad problem -- Sum involving many square roots....

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

    Determine the sum of the windage and friction losses for this motor

    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...
  42. E

    Verify that the sum of three quantities x, y, z

    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...
  43. moriheru

    Finding the maximum value of a sum involving several parameters

    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...
  44. D

    MHB How can i check if this sum is converges?

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

    MHB Why Does tan(x + pi/2) Equal -cotx in Trigonometry?

    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?
  46. N

    LEDs in parallel don't consume the sum of individual currents?

    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...
  47. yecko

    Sum to Infinity: Proving Divergence

    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...
  48. Adgorn

    Annihilator of a Direct Sum: Proving V0=U0⊕W0 for V=U⊕W

    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...
  49. T

    Number of pairs of integers satisfying this sum

    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\\ ##...
  50. Const@ntine

    Comp Sci 5x5 Arrays - Sum & Difference - (Fortran)

    Homework Statement Create two 5x5 arrays, A & B, and ask the person to fill them out. Save those numbers in matrix_a.txt & matrix_b.txt respectively. Then, save the sum and difference of those numbers in sum.txt & diff.txt respectively. Basically we need to create two arrays, fill them out...
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