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. arivero

    Sum of spin for chiral particles?

    How does the spin of a pair of particles work if both particles are known to be chiral? generically if I sum the spins of two different (EDIT: spin 1/2, indeed ;-) particles I expect to get a triplet with S=1 \uparrow\uparrow, \uparrow\downarrow+\downarrow\uparrow, \downarrow\downarrow and...
  2. F

    How many ways to write 4 as the sum of 5 non-negative integers?

    the problem: In how many ways can we write the number 4 as the sum of 5 non-negative integers?I realize this is a generalized combinations problem. I can plug it in using a formula, but I want to understand the logic behind why the generalizaed combination formula works. More specifically, my...
  3. C

    Steps to differentiate a geometric sum

    Can someone guide me with the steps to differentiate a geometric sum, x? ^{n}_{i=0}\sumx^{i}=\frac{1-x^{n+i}}{1-x} If I'm not wrong, the summation means: = x^0 + x^1 + x^2 + x^3 + ... + n^i Problem is: I have basic knowledge on differentiating a normal numbers but how do I apply...
  4. A

    Expressing signal as a sum of functions

    Homework Statement V(t) = [2t - 4] u(t) u(4 - t) I need to express this a sum of step and ramp functions. Homework Equations The Attempt at a Solution I have absolutely no idea how to proceed
  5. M

    MHB Probability of Rolling Sum of Die-Rolling

    I ran into this problem, and would like to see if there is something more elegant. Suppose we have a sequence $a_1, a_2, \dotsc, a_n, \dotsc$ where $a_k$ is the (running) sum of rolling a standard 6-side die $k$ times. E.g. What's the chance of saying the number $2$ appears in this sequence...
  6. mathbalarka

    MHB Double Sum Challenge: Equate the Limit

    Equate the limit $$\lim_{n \to \infty} \frac1{n} \sum_{i = 1}^n \sum_{j = 1}^n \frac1{i + j}$$ Note : This was a challenge from a user in mathstackexchange. From a glance, there should be many ways to do it, so partly I posed this problem to see how the resident analysts in MHB handle it...
  7. Dethrone

    MHB Minimize sum of number & reciprical + proof

    Find a positive number such that the sum of the number and its reciprocal is as small as possible. Full marks for proving your answer is correct. Process: let $a>0$ $$f(a)=a+\frac{1}{a}$$ $$f'(a)=1-\frac{1}{a^2}=\frac{(a+1)(a-1)}{a^2}$$ The critical numbers are $0$, $\pm 1$, but only $1$ is...
  8. anemone

    MHB Find the sum of all real numbers

    Find the sum of all real numbers $a$ such that $5a^4-10a^3+10a^2-5a-11=0$.
  9. T

    Algebraic manipulation of sum of squares.

    Hiya. I got to an interesting bit in a calculus book, but as usual I'm stumped by a (probably simple) algebraic step. The author goes from: (ds)^2=(dx)^2+(dy)^2 to: ds=\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx I understand moving the square root across, but I don't understand how the...
  10. N

    Rewriting bionomial sum using partial derivative

    Hi. Assume there's a probability ##q## for a guy to take a step to the right, and ##p=1-q## to take one to the left. Then the probability to take ##n## steps to the right out of ##N## trials is ##P(n) = {{N}\choose{n} }q^n p^{N-n}##. Now, what is ##<n>##? My textbook in statistical physics...
  11. anemone

    MHB Find Integer $k$ to Satisfy Sum of Inverse Progression > 2000

    Find an integer $k$ for which $\dfrac{1}{k}+\dfrac{1}{k+1}+\dfrac{1}{k+2}+\cdots+\dfrac{1}{k^2}>2000$.
  12. G

    Sum of singular 1-cubes = boundary of a singular 2-cube?

    Homework Statement I'm doing question 23 in Chapter 4 of Spivak's Calculus on Manifolds. The question asks, For R > O, and n an integer, define the singular l-cube, c_{R,n} :[0,1] \rightarrow \mathbb {R}^2 - 0 by c_{R,n} (t) = (Rcos2\pi nt, Rsin2\pi nt). Show that there is a singular...
  13. anemone

    MHB Prove a sum is not the fifth power of any integer

    Suppose $X$ is a number of the form $\displaystyle X=\sum_{k=1}^{60} \epsilon_k \cdot k^{k^k}$, where each $\epsilon_k$ is either 1 or -1. Prove that $X$ is not the fifth power of any integer.
  14. I

    MHB Magnitude and angle of vector sum

    help! find the magnitude of the resultant force and the angle it makes with the positive x-axis. i don't have any examples in my book like this one
  15. Math Amateur

    MHB Direct Sum of n Vector Spaces Over F - Knapp Proposition 2.31 - Pages 61-62

    I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding Theorem 2.31 (regarding the direct sum of n vector spaces) on pages 61-62. Theorem 2.31 and its accompanying text...
  16. Math Amateur

    MHB Universal Mapping Property of a Direct Sum - Knapp Pages 60-61

    I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding the Universal Mapping Property of direct sums of vector spaces as dealt with by Knapp of pages 60-61. I am not...
  17. I

    MHB What is the sum of the series?

    $\sum_{n=0}^{\infty}\frac{2^n}{3^nn!}$ is this correct? $\sum_{n=0}^{\infty}(\frac{2}{3})^n \frac{1}{n!}$ $\sum_{n=0}^{\infty}\frac{(x)^n}{n!}=e^x$ $x=2/3$ $e^x=e^{2/3}
  18. N

    How to rewrite the provided sum in another form?

    Homework Statement How to get from Sum of 2(cos((3pi)/(2^(k+1)))sin(pi/(2^(k+1)))) from k = 1 to infinity to Sum of sin((4pi)/(2^(k+1))) - sin((2pi)/(2^(k+1))) from k = 1 to infinity The two expressions are equivalent. I need help getting from the first expression to the second.
  19. anemone

    MHB Prove a sum is a composite number

    For positive integers $p,\,q,\,r,\,s$ such that $ps=q^2+qr+r^2$, prove that $p^2+q^2+r^2+s^2$ is a composite number.
  20. T

    Prove the Sum Rule for Limits

    Prove the Sum Rule for Limits $$\lim_{x\to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) = L + M$$ Proof Assume the following: $$\lim_{x \to a} f(x) = L, \space\lim_{x \to a} g(x) = M$$ Then, by definition ##\forall \epsilon_1 > 0, \exists \delta_1 > 0## such that...
  21. I

    MHB Find Sum of Series to Within 0.01

    how many term of the series $\sum_{n=2}^{\infty}\frac{1}{[n(ln (n))^2]}$ would you need to add to find its sum to within 0.01? approximate the sum of the series correct to four decimal places. $\sum_{n=1}^{\infty}\frac{(-1)^n}{3^nn!}$
  22. Y

    Determine the sum of the given series:

    Homework Statement Sum starting from n=1 to infinity for the expression, (3/4^(n-2)) What the solutions manual has done is multiply the numerator and the denominator by 4. 12/(4^(n-1)) I don't know what they have done from here on: 12 / (1 - 1/4) = 16 Can someone...
  23. A

    MHB Convergence of a Series: Is My Approximation Accurate Enough?

    Hey guys, I just wanted to run a quick series question by you guys just to confirm my answer. I'm doubting whether or not I should keep going or if S6 is enough. I got S5 = -0.28347 and S6 = -0.28347, so that is where I concluded than Sn ~ -0.2835. I would appreciate it if someone could...
  24. B

    MHB Sum Binomials: Proving Numerical Test Result

    I have this sum $$\left(N+1\right)^{2}\underset{j=1}{\overset{N}{\sum}}\frac{\left(-1\right)^{j}}{2j+1}\dbinom{N}{j}\dbinom{N+j}{j-1}\underset{i=1}{\overset{N}{\sum}}\frac{\left(-1\right)^{i}}{\left(2i+1\right)\left(i+j\right)}\dbinom{N}{i}\dbinom{N+i}{i-1}$$ and numerical test indicates that is...
  25. T

    Direct Sum and Direct Product: Understanding the Differences in Vector Spaces

    The definition (taken from Robert Gilmore's: Lie groups, Lie algebras, and some of their applications): We have two vector spaces V_1 and V_2 with bases \{e_i\} and \{f_i\}. A basis for the direct product space V_1\otimes V_2 can be taken as \{e_i\otimes f_j\}. So an element w of this space...
  26. anemone

    MHB Find the sum of all real solutions

    Find the sum of all real solutions for $x$ to the equation $\large (x^2+4x+6)^{{(x^2+4x+6)}^{(x^2+4x+6)}}=2014$. P.S. I know this doesn't count as a challenge(no matter how you slice it) because it's quite obvious and rather a very straightforward sort of problem but I'd like to share it...
  27. anemone

    MHB Find Sum of Diagonals of Pentagon $PQRST$

    Let $PQRST$ be a pentagon inscribed in a circle such that $PQ=RS=3$, $QR=ST=10$, and $PT=14$. The sum of the lengths of all diagonals of $PQRST$ equals to $\dfrac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
  28. J

    Proving the Induction Step: \sum_{j=1}^{k+2} j \cdot 2^j = (k+2)\cdot 2^{k+4}+2

    Homework Statement prove by induction \sum_{j=1}^{n+1} j \cdot 2^j = n \cdot 2^{n+2}+2; n \ge 02. The attempt at a solution P(0) \sum_{j=1}^{0+1} j \cdot 2^j = 0 \cdot 2^{0+2}+2 2+2 here is where I need some help is P(k) \sum_{j=1}^{k+1} j \cdot 2^j = (k+1) \cdot 2^{k+3}+2 ?? then...
  29. anemone

    MHB Sum Infinity Express: Rational Number Solution

    Express $\displaystyle \sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \dfrac{1}{m^2n+mn^2+2mn}$ as a rational number.
  30. J

    Finding the sum of 1^3 + 2^3 + + n^3 by induction

    1^3+2^3+...+n^3 = \left[ \frac{n(n+1)}{2}\right]^2; n\ge 1 P(1) = 1^3 = \frac{8}{8} = 1 P(k) = 1^3+...+k^3 = \left[ \frac{k(k+1)}{2}\right]^2 (induction hypothesis) P(k+1) = 1^3+...+k^3+(k+1)^3 = \left[\frac{(k+1)(k+2)}{2}\right]^2 I start getting stuck here I foiled it out then let m =...
  31. S

    MHB Sum of Prabhakar's 3 Param Mittag-Leffler

    I have a series in Prabhakar three parametric mittag-leffler is there any article on that
  32. J

    Write F as a sum of an orthogonal and parallel vector

    an object is moving in the direction i + j is being acted upon by the force vector 2i + j, express this force as the sum of a force in the direction of motion and a force perpendicular to the direction of motion. the parallel would be \hat{i}+\hat{j} and the orthogonal would be \hat{i} -...
  33. D

    Subdividing an integral into a sum of integrals over a given interval

    How does one prove the following: \int^{c}_{a} f\left(x\right)dx = \int^{b}_{a} f\left(x\right)dx +\int^{c}_{b} f\left(x\right)dx where f\left(x\right) is continuous in the interval x\in \left[a, b\right], and differentiable on x\in \left(a, b\right). My approach was the following...
  34. kaliprasad

    MHB Find Number b/w 1000-2000: Impossible Sum of Consec. Nums

    Find the number between 1000 and 2000 that cannot be expressed as sum of (that is >1) consecutive numbers.( To give example of sum of consecutive numbers 101 = 50 + 51 162 = 53 + 54 + 55 ) and show that it cannot be done
  35. Saitama

    MHB Evaluating a Sum Problem: Find Value

    Problem: Let $[x]$ be the nearest integer to $x$. (For $x=n+\frac{1}{2}, n\in \mathbb{N}$, let $[x]=n$). Find the value of $$\sum_{m=1}^{\infty} \frac{1}{[\sqrt{m}]^3}$$ Attempt: I tried writing down a few terms and saw that $1$ repeats $2$ times, $2$ repeats $4$ times but I didn't check it...
  36. kq6up

    Finding the Sum of an Infinite Series

    Homework Statement Find the expectation value of the Energy the Old Fashioned way from example 2.2. Homework Equations ##\left< E \right> =\frac { 480\hbar ^{ 2 } }{ \pi ^{ 4 }ma^{ 2 } } \sum _{ odds }^{ \infty }{ \frac { 1 }{ { n }^{ 4 } } } ## The Attempt at a Solution Never...
  37. anemone

    MHB Find the maximum of a sum

    If $p,\,q,\,r,\,s$ are positive integers with sum 63, what is the maximum value of $pq+qr+rs$?
  38. S

    Finding sum of infinite series

    Homework Statement Recognize the series $$3-3^3/3!+3^5/5!-3^7/7!$$ is a taylor series evaluated at a particular value of x. Find the sumHomework Equations Sum of Infinite series = ##a/1-x## The Attempt at a Solution So, I can't figure out what i would us as the ratio (the thing you multiply...
  39. D

    Proof question: the sum of the reciprocals of the primes diverges

    The gist of the approach I took is that∑1/p = log(e^∑1/p) = log(∏e^1/p) and logx→ ∞ as x→∞. Proof outline: let ∑1/p = s(x). (...SO I can write this easily on tablet) and note that e^s(x) diverges since e^1/p > 1 for any p and the infinite product where every term exceeds 1 is divergent. Then...
  40. J

    MHB Evaluation of Infinite sum of Inverse Trig. Series.

    How can we prove $$\displaystyle \tan^{-1}\left(\frac{4}{7}\right)+\tan^{-1}\left(\frac{4}{19}\right)+\tan^{-1}\left(\frac{4}{39}\right)+\tan^{-1}\left(\frac{4}{67}\right)+...\infty = \frac{\pi}{4}+\cot^{-1}(3)$$ My Trial: First we will calculate $\bf{n^{th}}$ terms of Given Series...
  41. Albert1

    MHB Finding the Sum of $w,x,y,z$ Given $2^w+2^x+2^y+2^z=20.625$

    if $2^w+2^x+2^y+2^z=20.625$ here $w>x>y>z$ and $w,x,y,z \in Z$ find $w+x+y+z$
  42. Albert1

    MHB Find the Min Sum of $m$ and $n$ for $(4^m+4^n)\ mod\ 100=0$

    if $(4^m+4^n)$ mod 100=0 (here $m,n\in N \,\, and \,\,m>n$) please find:$min(m+n)$
  43. J

    Vector Sum of a Standing Wave Confusion

    Hi, I was taught that a standing wave is formed when a progressive wave meets a boundary and is reflected. I was also taught that waves that meet a fixed end, reflect on the opposite side of the axis to the side that they met it at. (I hope that makes sense) If this is true, when the wave is...
  44. F

    Difference of a function and a finite sum

    Hi everybody, I am looking for some help with a problem that has been nagging me for some time now. I'm going to give you the gist of it, but I can provide more details if needed. So, after some calculations I am left with a function of the following form $$ F_L(y) = f(y) -S_L(y)...
  45. adjacent

    C# [C#] Sum of first x natural numbers

    I am writing this in C#. Here is the code. using System; namespace ConsoleApplication3 { class Program { static void Main(string[] args) { int sum = 0; int uservalue; Int32.TryParse(Console.ReadLine(),out uservalue)...
  46. Saitama

    MHB Efficient Methods for Evaluating Complex Sums: A Scientific Approach

    While doing an another problem, I came across the following sum and I have no idea about how one should go about evaluating it. $$\sum_{k=0}^{\infty} (-1)^k\left(\frac{1}{(3k+2)^2}-\frac{1}{(3k+1)^2}\right)$$ Wolfram Alpha gives $-\frac{2\pi^2}{27}$ as the result but I have absolutely no idea...
  47. DreamWeaver

    MHB Finite Binomial Sum: Proving 1 + 1/2 + 1/3 + ... + 1/n

    Show that \sum_{j=1}^{j=n}\binom{n}{j} \frac{(-1)^{j+1}}{j} = 1 +\frac{1}{2} +\frac{1}{3} + \cdots +\frac{1}{n}
  48. DreamWeaver

    MHB Sum of two inverse tangent functions

    By considering the product of complex numbers: z = (2+i)(3+i) Show that \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3} = \frac{\pi}{4}
  49. A

    What is the sum of infinite number of zeros?

    dear all, let's consider a length L and divide it into N number of small segments uniformly. Then the length of every segment should be L/N. then we add these segments up, which is L=\sum\frac{L}{N} then we take the limit N→\infty at both sides, this means...
  50. M

    Simple problems regarding sum of IID random variables

    Hi! I'm taking my first course in statistics and am hoping to get some intuition for this set of problems... Suppose I have a bowl of marbles that each weighs m_{marble}=0.01 kg. For each marble I swallow, there is a chance p=0.53 that it adds m_{marble} to my weight, and chance 1-p that...
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