Sum Definition and 1000 Threads

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. S

    Lie theory - sum of nilpotent ideals is nilpotent?

    I can't wrap my head around this proof that the sum of two nilpotent ideals is nilpotent, I get stuck at one stage: http://imageshack.com/a/img706/5732/5wgq.png I'm fine with every except showing by induction (I+J)^{N+k} = I^k \cap J + I \cap J^k . Here's my attempt; Base case: k =...
  2. S

    MHB How Can Complex Analysis Be Used to Sum Powers of Sine Functions?

    How to find the sum using complex analysis sin^3x+sin^32x+sin^33x+sin^34x+...+sin^3nx
  3. L

    MHB Optimization of the sum of the surfaces of a sphere and cube

    If the sum of the surfaces of a cube and a sphere as is constant, deierminar the minion of the diameter of the sphere to the edge of the cube in cases in which: 272) The sum of the volumes is minimal 273) The sum of the volumes is maximum And the answer are 272 = 1 and 273 = infinit Ok Vs =...
  4. A

    Evaluating a Finite Sum to a Closed Form Expression

    I have a finite sum of the form: ∑n=1Nexp(an+b√(n)) Is there any trick to evalute this sum to a closed form expression? e.g. like when a finite geometric series is evaluated in closed form.
  5. M

    Expressing sum of sines and cosines as a complex exponential

    If I'm given a function ##f(x) = A cos (x) + B sin (x)##, is there any way to turn this into an expression of the form ##F(x) = C e^{i(x + \phi)}##? I know how to use Euler's formula to turn this into ## \alpha e^{i(x + \phi)} + \beta e^{-i(x + \phi)}##, but is there a way to incorporate the...
  6. F

    MHB Answer: Image Direct Sum & Linear Operator: Is Union Equal?

    Given 2 subspaces and a linear operator, is the image of the direct sum of the subspaces equal to the union of the images under the operator? Thanks
  7. A

    How can you approximate a sum by an integral without having a small Δx?

    The attached pdf shows integral approximations of two sums, which are done in my book. In the first there is no result but the book simply states that one can approximate the sum by an integral. My question is: How is this done? Normally when you approximate a sum by an integral you have a sum...
  8. N

    Sum of independent exponential distributions with different parameters

    Homework Statement As the title indicates. I'm given two independent exponential distributions with means of 10 and 20. I need to calculate the probability that the sum of a point from each of the distributions is greater than 30. Homework Equations X is Exp(10) Y is Exp(20) f(x) =...
  9. K

    Finding the sum vector given only the moduli and angle

    Homework Statement Given two 2-dimensional vectors \overline{a} and \overline{b} of moduli l\overline{a}l = 3u and l\overline{b}l = 4u, and forming an angle  of 120 degrees between them, determine the modulus of the sum vector \overline{s} = \overline{a} + \overline{b} and the angle between...
  10. 9

    Sequence of 3/4^(2k). Show is convergent, find sum

    Homework Statement Consider the sequence ak = (3)/(4^(2k)). Show is convergent, find sum. Please check work. Homework Equations ak = 3/(4^2k) let s {n} be the series associated with the sequence. Cannot write summation notation here, but k starts at 1 (k = 1 on bottom) and infinity...
  11. L

    How can ln be used to solve a sum to infinity problem?

    Homework Statement Prove the following result: \frac{1}{2.2} + \frac{ 1}{3.2^2} + \frac{1}{4.2^3} ... = 2ln2 -1 Homework Equations The Attempt at a SolutionI tried writing down the nth term of the series which is 1/(n+1)2^n But don't know where to move after this.
  12. N

    Finding the PDF of the Sum of Two Random Variables: Uniform Distribution

    Homework Statement X is uniform [e,f] and Y is uniform [g,h] find the pdf of Z=X+Y Homework Equations f_z (t) = f_x (x) f_y (t-x) ie convolution The Attempt at a Solution Obviously the lower pound is e+g and the upper bound is f+h so it is a triangle from e+g to f+h...
  13. L

    What is the solution to the infinite series involving factorials and pi?

    Homework Statement Sum to infinity \frac{1}{2!} - \frac{ \pi ^2}{4^2.4!} + \frac{\pi^4}{4^4.6!} ... Homework Equations The Attempt at a Solution I thought the series was similar to the Maclaurin expansion of cos x so I tried putting in x= ∏/4 But I end up with the...
  14. L

    Prove: Sum of Series Equals e^2 - e

    Homework Statement Prove the following result: \frac{1}{1!} + \frac{1+2}{2!} + \frac {1+2+2^2}{3!} ... = e2 - e Homework Equations The Attempt at a Solution Could someone please give me a hint on what to do . I tried writing out the maclaurin series for e^2 and e but...
  15. B

    Can the Sum of Natural Numbers Really Be -1/12?

    A professor in Cambridge University has showed his proof that the sum of natural number was equal to -1/12. The video can be found on the internet.Well, although the way to proof I think is really ridiculous, it could be a good way to building a new math model. Since we think the sum of natural...
  16. N

    The sum of two vectors, A→ + B→

    Homework Statement The sum of two vectors, A→ + B→, is perpendicular to their difference, A→ - B→. How do the vectors magnitude compare? The Attempt at a Solution SQRT[(A+B)^2 + (A-B)^2]
  17. anemone

    MHB Sum of two trigonometric terms

    Prove that $\tan \left( \dfrac{3 \pi}{11} \right)+ 4\sin \left( \dfrac{2 \pi}{11} \right)=\sqrt{11}$. I know this problem may be stale as it has been posted countless times at other math forums, but I've seen one brilliant method to attack this problem recently, and I'm so eager to share it...
  18. Albert1

    MHB Finding the Min Sum of a Sequence of Numbers in $Z$

    $x_0,x_1,-----,x_{2004} \in Z , \, x_0=0, \mid x_n \mid =\mid x_{n-1}+1\mid $ $for, \,\, 1 \leq n \leq 2004$ (1) $find :\,\, min\mid x_1+x_2+x_3+ ------+x_{2004}\mid $ (2) get a set of numbers $ x_1,x_2,-----x_{2004} $ satisfying your answer
  19. anemone

    MHB Evaluating the Sum of $a+b+c$ for Equation $x^5-12x^4+ax^3+bx^2+cx-64=0$

    Hi MHB, I have solved the problem as stated below but I don't know if it's an unique solution and even if it is, I have no idea how to prove that would be the case. Can anyone show me how to approach the problem correctly? For the equation $x^5-12x^4+ax^3+bx^2+cx-64=0$, all of its roots are...
  20. M

    Least amount of terms for sum.

    I was trying to design some GUI for a tool I'm making and I noticed there's a hidden math problem somewhere in there. Not being one to let the opportunity slide, I decided it's worth exploring. Basically there's 3 buttons that add to a variable. What are the best values to put on those...
  21. A

    The Sum of All the Natural Numbers

    Hi lovely people, I recently came across a video http://www.youtube.com/watch?v=w-I6XTVZXww that said if you add all of the natural numbers from 1 to infinity, the answer is... What do you think it is? Infinity or something like that? They said it was -1/12. I watched the proof but I don't...
  22. Square1

    When sum of remainders becomes divisible

    Hi. I need to sort out some concepts and terminology. I was wondering if there are algorithms and terminology surrounding the following situation. Lets say I want to buy some object for an amount of money, but the object cost less than the amount of money I have. I will have a remainder of...
  23. L

    Sum of 1 + 2a + 3a2 + ... to n terms

    Homework Statement Sum the series 1 + 2a + 3a2 + ... to n terms This series consists of an a.p. (with general term n) and gp general term a^(n-1) right? So the series general term is na^(n-1) So is the sum the sum of each progression times each other? i.e (1-a^n)/(1-a) *...
  24. L

    Sum to infinity question (G.P.)

    Homework Statement ΔABC has AB= 8 in, BC= 10in , CA= 6in . AD is the perpendicular from A to BC. DE the perpendicular from D to AB, EF the perpendicular from E to BD and so on. Show that CA + AD + DE+... is a geometric series and find it's sum to infinity. Homework Equations The...
  25. anemone

    MHB Compute a square root of a sum of two numbers

    Compute $\sqrt{2000(2007)(2008)(2015)+784}$ without the help of calculator.
  26. anemone

    MHB Find Sum of $\dfrac{4k}{4k^4+1}$

    Determine the sum \sum_{k=1}^n \dfrac{4k}{4k^4+1}.
  27. L

    Sum of IID random variables and MGF of normal distribution

    If the distribution of a sum of N iid random variables tends to the normal distribution as n tends to infinity, shouldn't the MGF of all random variables raised to the Nth power tend to the MGF of the normal distribution? I tried to do this with the sum of bernouli variables and...
  28. mente oscura

    MHB Express numbers, raised to an exponent, for sum of consecutive numbers.

    Hello. Not long ago, I did a study on numbers, raised to an exponent. I noticed that a "pattern" remained, and I could find a general formula. Let: a , n, k \in{N}, when "a" is odd number: I define: a _1, \ldots , a_k \in{N}, as the consecutive addends, such that: Let: a , n, k \in{N}...
  29. anemone

    MHB Find the sum of the first n terms

    Evaluate the sum $\displaystyle \sum_{i=0}^n \tan^{-1} \dfrac{1}{i^2+i+1}$.
  30. anemone

    MHB Evaluate the sum of the reciprocals

    Given $p+q+r+s=0$ $pqrs=1$ $p^3+q^3+r^3+s^3=1983$ Evaluate $\dfrac{1}{p}+\dfrac{1}{q}+\dfrac{1}{r}+\dfrac{1}{s}$.
  31. NATURE.M

    Proving the Riemann Sum for the Integral of x^2 from 1 to 3

    So my textbook asks to show \int^{3}_{1} x^{2}dx = \frac{26}{3}. They let the partition P = {x_{0},...,x_{n}}, and define the upper Riemann sum as U(P) = \sum^{i=1}_{n} x_{i}Δx_{i} and lower sum as L(P) = \sum^{i=1}_{n} x_{i-1}Δx_{i} I understand this part, but the next part is where I'm...
  32. I

    Solving for a factor in a large sum

    Dear everyone. First of all Merry Xmas, when everybody gets to that. I have a problem solving for a factor within a sum. My formula looks as follows: T = Æ© It * A0t The sum runs from t=1 to N, and the aim is to solve for A0, but all my calculations end up extremely messy...
  33. anemone

    MHB Evaluate Sum: $\cos^3\beta/\cos\alpha + \sin^3\beta/\sin\alpha$

    If $\dfrac{\cos \alpha}{\cos \beta}+\dfrac{\sin \alpha}{\sin \beta}=-1$, evaluate $\dfrac{\cos^3 \beta}{\cos \alpha}+\dfrac{\sin^3 \beta}{\sin \alpha}$.
  34. P

    MHB Unit sum composed of unit fractions

    Is it necessary for a unit sum composed of unit fractions to include 1/2? Doing maple runs this seems to be the case, but it is not evident to me how this could be Edit: In fact it seems it could not be, given the Erdos Graham problem Erd?s?Graham problem - Wikipedia, the free encyclopedia But...
  35. W

    Sum of all possible products of elements taken from couples

    Hello I have N couples of real numbers higher than 1. Let's call them like (a0,b0), (a1,b1),...,(aN,bN) I have a number R <= N. I need the sum of all the possible products of N elements, chosing one from each couple but exactly R times the "b" element and N-R times the "a" element...
  36. P

    Sum of two independent uniform random variables

    Hi, http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter7.pdf (see page 8, sum of two independent random variables). I don't understand why they had to go further into the limits, 1 < z < 2. Why do they have to do that? And also, where did they get it...
  37. S

    MHB Evaluate the sum (1) ( 2 problems )

    Evaluate the sum \sum_{i=2}^{99}((i + 1)^2 - i^2)) So I found the pattern and got ((3^2) - 2^2)) + ((4)^2 - (3)^2)) + ((5^2) - (4)^2) ... etc etc 100^2 -2^2 = 9,996? Is this correct? #2 Evaluate the sum \sum_{i=2}^{100}(i^2 -(i - 2)^2) and got: 100^2 + 99^2 - 1 = 19,800. Is this correct?
  38. M

    What is the connection between linear and angular momentum?

    Homework Statement problem in in the picture Homework Equations I can't understand some parts of the equation. The Attempt at a Solution
  39. K

    MHB 'The sum of a series' question

    Assume that \sum_{n=1}^{\infty} a^2_{n} converge, and assume that a_{n} is non-negative for all \textit{n} \in N. Determine whether the following statement is true (and prove it) or false (and give counterexample). \sum_{n=2}^{\infty} \frac{a_{n}}{n^{2/3}}<\infty Does anyone know how to do...
  40. T

    Infinite product converges if and only if sum converges

    Homework Statement a_n is a sequence of positive numbers. Prove that \prod_{n=1}^{\infty} (1+a_n) converges if and only if \sum_{n=1}^{\infty} a_n converges. Homework Equations The Attempt at a Solution I first tried writing out a partial product: \prod_{n=1}^{N} (1+a_n) =...
  41. anemone

    MHB Evaluate the sum of a function

    Evaluate $h\left( \dfrac{1}{401} \right)+h\left( \dfrac{2}{401} \right)+\cdots+h\left( \dfrac{400}{401} \right)$ if $h(x)=\dfrac{9^x}{9^x+3}$.
  42. MarkFL

    MHB Minimize Sum of Line Segments Length w/ Point P on Line AD - Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can see my work.
  43. W

    Sum of all possible products when each product has a maximum

    Hello I have a set of sets of real numbers greater than 1. Each set can have a different quantity of numbers. Set A1 {a11, a12,...a1m1} Set A2 {a21, a22, ..., a2m2} ... Set AN {aN1, aN2, ..., aNmN} If I want the sum of all possible products that have one element from each set, that's...
  44. anemone

    MHB What is the sum of real roots for $q^4-7q^3+14q^2-14q+4=0$?

    Determine the sum of real roots of the equation $q^4-7q^3+14q^2-14q+4=0$.
  45. M

    How Many Combinations of Digits 0-7 Sum to 7?

    When you have combinations where digits are 0,1,2...,m, meaning we have n=m+1 and k, is there a way to see how much of them sum up to a given number? For the sake of simplicity I have the digits 0,1,2...,7 (so n=8), and k=3. I need to find how much of these combinations WITH repetition sum up to...
  46. Q

    Displacement Using Tabular Reimann Sum

    Homework Statement Find the displacement using a midpoint Reimann sum with 4 subintervals of equal width. minute 0, 12, 24, 36, 48, 60, 72, 84, 96 ft/min -4, -4, 1, 4, 5, -6, 0, 5, 2 Homework Equations Displacement is the final position minus starting position. The Attempt at...
  47. Q

    Solving Sum HW Trouble: b-a=2, n=4, delta x=1/2

    Homework Statement I'm having trouble writing a sum for this: https://scontent-b-mia.xx.fbcdn.net/hphotos-frc3/v/1420232_10201136345792485_382911431_n.jpg?oh=ebd9432103184956c863370121e326ce&oe=529332ED Homework Equations delta x = (b-a)/n n = 4 b-a = 2 delta x = 1/2 The...
  48. Q

    Is My Reimann Sum Solution Correct for 6/(1+2x) from 0 to 2 with 4 Subintervals?

    Homework Statement Approximate the value of the integral of 6/(1+2x) with respect to x from 0 to 2. Use 4 subintervals of equal width and use the left endpoints. Homework Equations delta x = (b-a)/N The Attempt at a Solution The integral is the sum of 6/(1+i) from i = 0 to i = N-1 or 3...
  49. Q

    Expressing Integral as Reimann Sum

    Homework Statement Express the integral of (sinx + 1) dx over the interval [0,pi] with a Reimann Sum using 4 subintervals of equal width and letting x_i^* be the left endpoint of the subinterval [x_(i-1), x_i] Homework Equations Δx = [b-a] / n The Attempt at a Solution Δx = pi/4...
  50. F

    Measuring sum of two components of spin angular momentum

    Homework Statement What are the possible results of the measurement of the sum of the x and z components of the spin angular momentum of a spin-1/2 particle? Sx = Spin angular momentum operator x Sz = Spin angular momentum operator x Homework Equations The Attempt at a...
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