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. J

    Is the Partial Sum Negative? Exploring Cosine Telescoping Series

    Homework Statement If you sum this from one to infinity. Ʃ (cos(1/(n)^2 - cos(1/(n+1)^2) Homework Equations The Attempt at a Solution Ʃ (cos(1/(n-1)^2 - cos(1/(n+1)^2) This is telescoping if you work that out for the partial nth partial sum you get cos(1) -...
  2. MarkFL

    MHB Realguy's question at Yahoo Answers regarding a Riemann sum

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  3. Q

    Reimann Sum Limit Homework Solution | Integration Limits and Delta X Formula

    Homework Statement https://scontent-b-mia.xx.fbcdn.net/hphotos-prn2/v/1419845_10201044047645089_1286462043_n.jpg?oh=adc74f67f112c0697cbfba79b4fa81fc&oe=5283F9AB Homework Equations delta x = (b-a)/n The Attempt at a Solution Well, from the delta x formula I can figure out the...
  4. Q

    What is the Riemann Sum Approximation for this Homework Problem?

    Homework Statement https://scontent-b-mia.xx.fbcdn.net/hphotos-prn2/v/1456973_10201043975243279_1765184125_n.jpg?oh=05b39611ad70d28d837ed219e1b0f2aa&oe=52838593 Homework Equations The area can be approximated by using the sum of the areas of the rectangles. Area of rectangle = change...
  5. Seydlitz

    Proof that the sum of complex roots are 0

    Homework Statement Hello guys, I need to prove that the sum of complex roots are 0. In the Boas book, it is actually written 'show that the sum of the n nth roots of any complex number is 0.' I believe it's equivalent. The Attempt at a Solution I have managed to obtain this summation. It is...
  6. D

    MHB Integral = 2pi sum res UHP + pi i sum res real axis

    \(\DeclareMathOperator{\Ima}{Im}\) \(\DeclareMathOperator{\Res}{Res}\) Given \[ \Ima\left[\int_{-\infty}^{\infty}\frac{e^{iz}}{z(\pi^2 - z^2)}dz\right]. \] I know the integral is equal to \[ 2\pi i\sum_{\text{UHP}}\Res(f(z); z_j) + \pi i\sum_{\mathbb{R}\text{ axis}}\Res(f(z); z_k). \] However...
  7. M

    Rewriting sum of iterated integrals (order of integration)

    Homework Statement Rewrite the given sum of iterated integrals as a single iterated integral by reversing the order of integration, and evaluate. $$\int_0^1 \int_0^x sin x dy dx + \int_1^2 \int_0^{2 - x} sin x dy dx$$ Homework Equations None The Attempt at a Solution I drew the domains of...
  8. N

    What is the smallest possible sum for a labeled 63-gon?

    Hello. When I was in Mathematical Olympiad in our district, I got a example which I can not solve :-( Now, I am very interested in, how to do it. I don't know, my brother - teacher don't know, my friends don't know. I think you can solve it :-) Can you help me? It isn't my homework and I...
  9. A

    MHB Condition Number of sum of Matrices

    As far as I know there is no explicit formulas but is this true? I've tested it in Matlab with random matrices and It seems true! cond(A+B) =< cond(A) + cond(B) Where can I find a proof for this hypothesis?
  10. countzander

    Marginal PDFs for Joint PDF of X and Y

    Homework Statement Suppose that ∫X,Y(x,y) = λ2e-λ(x+y), 0 ≤ x, 0 ≤ y Find E(X + Y) Homework Equations E(X + Y) = E(X) + E(Y) The Attempt at a Solution Since the expected vale of a sum is the sum of the expected values, I attempted to find the marginal pdfs of the joint pdf...
  11. M

    Sum of two closed subspaces in a Banach space

    Homework Statement . Let ##E## be a Banach space and let ##S,T \subset E## two closed subspaces. Prove that if dim## T< \infty##, then ##S+T## is also closed. The attempt at a solution. To prove that ##S+T## is closed I have to show that if ##x## is a limit point of ##S+T##, then ##x \in...
  12. K

    How to Take the sum of moments

    How to Take the sum of moments ! Homework Statement Hi i am doing static equilibrium exercises in textbook. Homework Equations But I don't know how to find sum of moments at P when its come to angle. This is a drawing The Attempt at a Solution I have No idea ! I was...
  13. J

    Is particle more than the sum of its fields?

    Electron is a charge, what means concrete configuration of electric field, growing to infinity in the center (if it would be a point particle). It is also a magnetic dipole moment - concrete configuration of magnetic field growing to infinity. And a mass - concrete configuration of gravitational...
  14. Gh778

    How to calculate sum of torque ?

    I would like to calculate sum of torque on object composed of one container of helium II liquid and one container of gas with low pressure. I know you think it's 0, but I would like to be sure (I think it's not 0, but it's only my intuition). It's not a theoretical problem, so I take in account...
  15. M

    MHB Calculating a Sum in C: Finding the Error in Floating Point Precision?

    Hi! I want to write a program in c, that calculates the sum [1+sum{1/(i(i+1)) from 1 to n}]. I declared the variables as float. When I run the program with n=9, the output is 1.899999976, but when I calculate this with a calculator the result is 1.9. Where is the error?
  16. K

    Probability of getting a sum of 13

    Obtain the probability of getting a sum of 13, when four fair dice are rolled together once. if we do by just calculating all possible values of sum,then it will take more time. so,we can do above problem as Multinomial Coefficents of sum i.e.,x1+x2+x3+x4 = 13 where,1<= xi <= 6,for all 1<= i <=...
  17. O

    Can You Find the Best Constant for Sum-Free Subsets?

    A set A of non-zero integers is called sum-free if for all choices of a,b\in A, a+b is not contained in A. The Challenge: Find a constant c > 0 such that for every finite set of integers B not containing 0, there is a subset A of B such that A is sum-free and |A| ≥ c|B|, where |A| means the...
  18. E

    Sum or Integral? Understanding the Calculation of Casimir Effect Measurements

    I am reading the paper on the Casimir effect and they measure the space in between the plates using a sum and the energy of the vacuum without the plates using an intregal. Why do they use the sum and intregal, should they be switched.
  19. Sudharaka

    MHB Direct Sum Property: Proving Uniqueness

    Hi everyone, :) I encountered this question and thought about it several hours. I am writing down my answer. I would greatly appreciate if somebody could find a fault in my answer or else confirm it is correct. :) Problem: Let \(V_1,\,\cdots,\,V_k\) be subspaces in a vector space \(V\)...
  20. Superposed_Cat

    Can a Sigma Sum be Integrated with Fubini's Theorem?

    How would you Integrate a Sigma Sum?
  21. Seydlitz

    Using Table and Computer to find the sum of a series

    I'm doing problem section 15 chapter 1 by Boas. I don't want to ask about a particular problem in there but she often gives this kind of instruction, "By computer or tables, find the exact sum of each of the following series." My question is, what kind of table she is referring to? I take it...
  22. ajayguhan

    Why Does Sum of Matrix Elements x Cofactor in Different Rows Equal Zero?

    In a matrix the sum of element of a matrix in a row times it's co factor of that elemt gives the determinant value, but why does the sum of element of a matrix times cofactor of different row is always zero?
  23. D

    Show that the natural representation of S3 is a direct sum of irreps

    Homework Statement Hey everyone! So to elaborate the title a bit more: basically I have to show that the natural representation of S_{3} is a direct sum of the one-dimensional irreducible representation and the two-dimensional irreducible representation of S_{3}. Homework Equations Im...
  24. anemone

    MHB Find the sum of the first 11 terms of given series

    Hi MHB, This problem vexes me until my mind hurts. Problem: Find the sum of the first 11 terms of the series \frac{19}{99}+\frac{199}{999}+\frac{1999}{9999}+ \cdots Attempt: I managed only to find the expression of the nth term of the given series and I got...
  25. R

    Probability that sum of two random variables is greater than 1

    Homework Statement Let us choose at random a point from the interval (0,1) and let the random variable X_1 be equal to the number which corresponds to that point. Then choose a point at random from the interval (0,x_1), where x_1 is the experimental value of X_1; and let the random variable...
  26. O

    Sum of 1/(n^2) as n goes to infinity

    Homework Statement Prove Ʃ1/(n^2) as n goes to infinity = (∏^2)/8 Homework Equations The Attempt at a Solution No idea how to start. Pls guide. Thanks
  27. L

    Interchanging Linear Operator and Infinite Sum

    Suppose that x\in H, where H is a Hilbert space. Then x has an orthogonal decomposition x = \sum_{i=0}^\infty x_i. I have a linear operator P (more specifically a projection operator), and I want to write: P(x) = \sum_{i=0}^\infty P(x_i). How can I justify taking the operator inside the...
  28. anemone

    MHB Sum and Product of Real Roots of A Quartic Function

    Let $p$ be the sum and $q$ be the product of all real roots of the equation $x^4-x^3-1=0$. Prove that $q<-\dfrac{11}{10}$ and $p>\dfrac{6}{11}$.
  29. alyafey22

    MHB What is the Interesting Euler Sum Proven by this Equation?

    Prove the following Euler sum \sum_{k\geq 1}\left(1+\frac{1}{3}+\cdots +\frac{1}{2k-1} \right) \frac{x^{2k}}{k}=\frac{1}{4}\ln^2\left( \frac{1+x}{1-x}\right)
  30. D

    Fourier coefficients and partial sum of Fejer

    Homework Statement f(t) a continuously differentiable function twice over the circle T1 cr its Fourier coefficients and σn(f,t) partial sum of Fejer. a.Demonstrate that http://imageshack.us/a/img94/5992/ds35.png b. Consider k as -n≤k≤n , using cr coefficients calculate...
  31. O

    MHB Proving that the sum of 2 measurable functions is measurable

    I know there are many proofs for this but I am having trouble proving this fact using my book's definition. My book defines first a non negative measurable function f as a function that can be written as the limit of a non decreasing sequence of non-negative simple functions. Then my book...
  32. S

    MHB Sum of Products with Karnaugh Map

    Write out the minimal Sum of Products(SOP) equation given the following Karnaugh Map. YZ|WX 00 01 11 10 00 d 1 1 1 01 1 1 0 0 11 0 0 d 1 10 0 0 0 0 Need someone to check my answer. My answer: yzw + \bar{y}\bar{z} + \bar{y}\bar{w}
  33. N

    Integral: square root of sum of trig polynomials

    Hi, I am trying to make progress on the following integral I = \int_0^{2\pi} \sqrt{(1+\sum_{n=1}^N \alpha_n e^{-inx})(1+\sum_{n=1}^N \alpha_n^* e^{inx})} \ dx where * denotes complex conjugate and the Fourier coefficients \alpha_n are constant complex coefficients, and unspecified...
  34. D

    Is V a Direct Sum of V+ and V-?

    Homework Statement Let ##T\in L(V,V)## such that ##T^{2}=1##. Show that ##V=V_{+}\oplus V_{-}## where ##V_{+}=\{v\in V:T(v)=v\}## and ##V_{-}=\{v\in V:T(v)=-v\}##.The Attempt at a Solution I was given a theorem that said that ##V## is the direct sum if and only if every vector in ##V## can be...
  35. Gh778

    Sum of forces, vacuum and gravity

    It's a theoretical study. I would like to understand how the sum of forces can be at 0 if I put an object (vacuum in it) in a big liquid disk (disk is fulled with liquid), the disk is big enough for agglomerate liquid (like this works with a planet, matter is agglomarate with gravity). There is...
  36. anemone

    MHB Can $3^{2008}+4^{2009}$ Be Factored into Two Numbers Larger Than $2009^{182}$?

    Show that $3^{2008}+4^{2009}$ can be written as product of two positive integers each of which is larger than $2009^{182}$
  37. M

    Sum of independent Random Variables

    Homework Statement Three yearly losses. First: Exponential Second & Third: Weibull Losses are independent. Find the 95% VaR of the min loss Homework Equations The Attempt at a Solution My first thought was: Let L be total loss, A be first Loss, B be second loss, C be third...
  38. F

    Finding a of n from Sn partial sum

    Homework Statement suppose that the partial sum of the series (sigma)n=1,infinity an is given by the partial sum Sn = (-2n+9)/(-6n+15). Find an expression for an when n>1 Homework Equations Sn= (-2n+9)/(6n+15 The Attempt at a Solution So I attempted to subtract S(n-1) from S(n) to get each...
  39. C

    MHB Sum series- Prove the equality of ratio and root.

    I found this on the internet, but did not find the proof. Curious to me is that the the ratio and root test have the same conditions. How can i basically prove this equality? \frac{a_{n+1}}{a_{n}} = \sqrt[n]{a_{n}} Thank you!
  40. hxthanh

    MHB What is the result of the sum of binomial coefficients with alternating signs?

    Evaluate sum: $\displaystyle S=\sum_{k=0}^{2n}(-1)^k{2n\choose k}{4n\choose 2k}$
  41. C

    MHB Sum series- convergence and divergence

    converge or diverge? \sum_{n=1}^{^{\infty }}a_{n} a_{1}= \frac{1}{3}, a_{n+1}= \sqrt[n]{a_{n}} Im having problems to solve this exercise, i would like to see your solutions
  42. anemone

    MHB Can you generalize the result for this sum over sum problem?

    Simplify \frac{\sum\limits_{k=1}^{99}\sqrt{10+\sqrt{k}}}{ \sum\limits_{k=1}^{99}\sqrt{10-\sqrt{k}}}
  43. hxthanh

    MHB How to Calculate the Value of a Given Sum in Mathematics?

    Put $1\le n\in\mathbb Z$ Find the Sum: $S_n=\displaystyle \sum_{k=1}^n\dfrac{2k+1-n}{(k+1)^2(n-k)^2+1}$
  44. polygamma

    MHB A sum involving the central binomial coefficients

    Wolfram MathWorld states that $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{3}}{18} \Big[ \psi_{1} \left(\frac{1}{3} \right) - \psi_{1} \left(\frac{2}{3} \right) \Big]- \frac{4}{3} \zeta(3) $$ where $\psi_{1}(x)$ is the trigamma function. But I can't get my answer in...
  45. J

    MHB Sum of 5 digit number(II)

    The Total Sum of $5$ Digit no. which can be formed with the Digit $0,0,1,1,2,2,3,3,4,4,4,5,6$ [a] without Repetition of Digit. [b] with Repetition of Digit
  46. J

    MHB Calculating the Total Sum of 5-Digit Numbers with and without Repetition

    The Total Sum of $5$ Digit no. which can be formed with the Digit $0,1,2,3,4,5,6,7$. [a] when repetition of digit is allowed [b] when repetition of digit is not allowed
  47. O

    Blanking on word for kind of convergence of a sum

    I have a sum \sum_{n=-\infty}^{\infty} f(n) which I do not want to consider the convergence of in the normal sense, but I want to talk about the limit \lim_{N\to \infty} \sum_{n=-N}^{N} f(n). I know that when this limit exists the sum is _____ convergent, or is a _____ sum, where _____ is...
  48. MarkFL

    MHB Have Sum Fun - Compute S_n

    Please compute the following sum: S_n=\sum_{k=1}^{n}\frac{n!}{(k-1)!(n-k)!}
  49. B

    Algebraic properites of the direct sum

    Is the following statement true? I am trying to see if I can use it as a lemma for a larger proof: Let ##V## be a vector space and let ##W, W_{1},W_{2}...W_{k} ## be subspaces of ##V##. Suppose that ## W_{1} \bigoplus W_{2} \bigoplus ... \bigoplus W_{k} = W ## Then is it always the case that...
  50. S

    Stats: Need help understanding where the sum of x^2 comes from

    Homework Statement I have no idea where the sum of x^2 comes from, from the information I posted. I know it must be something pretty simple but its completely going over my head. In the picture that I've attached, I am wondering where the 2431.72, 4901.66, and 3252.44 come from. Thank you...
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