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. yuming

    Telescopic sum issues, cant get Sk

    1. Homework Statement Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges Homework Equations 3/(1*2*3) + 3,/(2*3*4) + 3/(3*4*5) +...+ 3/n(n+1)(n+2) The Attempt at a Solution the first try, i tried using partial fraction which equals...
  2. karush

    MHB Dc 8t14 product to sum indentity

    4Use the power to sum formula to simplify the expression $\frac{\sin\left({3\theta}\right)+\sin\left({5\theta}\right)} {\cos\left({3\theta}\right)+\cos\left({5\theta}\right)}$ The answer is $\tan(4\theta)$ $$\sin\left({3\theta}\right)+\sin\left({5\theta}\right)...
  3. M

    Can one sum the polarizability of atoms?

    I'm interested in finding the polarizability of NaCl. Let's say I have the polarizability of ##Na^+## and of ##Cl^-##. Can I sum these to get the polarizability of NaCl, that is: $$\alpha_{NaCl} = \alpha_{Na^+} + \alpha_{Cl^-}$$ Honestly, that does seem very accurate, but perhaps it's to a...
  4. T

    Pascal - sum of digits in binary/hexadecimal

    Homework Statement Read two integers. First one tells you the type of your numeral system (binary, decimal, hexadecimal) the second one will be your number in decimal. Using functions or procedures I need to convert the number into the required system and then count the sum of its digits in...
  5. Albert1

    MHB Find the Sum of k when $k\in N$ and $\sqrt {k^2+48k} \in N$

    $k\in N$ , and $\sqrt {k^2+48k} $ $\in N$ find $\sum k$
  6. W

    Evaluating Sum from j=1 to n: i=k

    Homework Statement Evaluate ∑##_{j=1}^{n} \delta_{ij} \delta_{jk}## where 1≤i≤n and 1≤k≤n and ##\delta_{ij}## and ##\delta_{jk}## = 1 if i=j 0 otherwise Homework EquationsThe Attempt at a Solution I'm pretty unsure how to do this. I assume k and i are constant. If that's the case, wouldn't...
  7. M

    Supertrace Sum Rule and Fermi Constant in Electroweak Standard Model

    Two years ago, this paper appeared: http://arxiv.org/abs/1305.4208 Relation between masses of particles and the Fermi constant in the electroweak Standard Model G. Lopez Castro, J. Pestieau (Submitted on 17 May 2013) An empirical formula relating the physical masses of elementary particles and...
  8. J

    Sum of Translational and Angular Forces

    Hello, I'm trying to figure out the free body diagrams for the inverted pendulum problem and I'm having trouble figuring out the one equation: ##Psin\theta + Ncos\theta - mgsin\theta = ml\ddot{\theta} + m\ddot{x}cos\theta## I've never really seen a mixed sum of forces equation before where some...
  9. G

    Use of binomial theorem in a sum of binomial coefficients?

    Homework Statement How to use binomial theorem for finding sums with binomial coefficients? Example: S={n\choose 1}-3{n\choose 3}+9{n\choose 5}-... How to represent this sum using \sum\limits notation (with binomial theorem)? Homework Equations (a+b)^n=\sum\limits_{k=0}^{n}{n\choose...
  10. Sarah00

    Finding the Sum of an Alternating Geometric Sequence

    Hi! If I have a sequence that its first 4 terms are: 30, -31, +32, -32 The pattern is geometric sequence but has alternating signs.. How can I find its sum .. I know it is composed of 2 sequences .. However, when I try to separate the 2 sequences .. I get them of different "lengths" In...
  11. G

    Mean of the square of a sum of exponential terms

    Homework Statement [/B] Calculate \widehat{Y^{2}} (i.e., the mean of the square of Y. Homework Equations Y=\sum_{k=0}^{N-1}y_{k} where y_{k}=e^{-\gamma t}e^{\gamma \tau k}G_{k} and t=N\tau The quantities y_{k} (or G_{k}) are statistically independent. The Attempt at a Solution...
  12. C

    MHB Using Reimann sum to estimate the value of a double integral

    If R = [−3, 1] × [−2, 0], Use a Riemann sum with m = 4, n = 2 to estimate the value of ∫∫R(y2 − 2x2) dA. Take the sample points to be the upper left corners of the squares. So far, I found the indefinite integral of the function to be y3/3 - 2x3/3 Not sure where to go from here
  13. wirefree

    Understanding Complex Exponential Summation: How is the Arctan Function Used?

    I appreciate the opportunity afforded by this forum to submit a question. I have struggled with the derivation shown in the attached picture. I am certainly unfamiliar with the concept used to include the arctan function in the encircled step. Would be highly appreciative of a prompt.wirefree
  14. B

    Convergence of a sum over primes

    I am trying to understand a condition for a nonincreasing sequence to converge when summed over its prime indices. The claim is that, given a_n a nonincreasing sequence of positive numbers, then \sum_{p}a_p converges if and only if \sum_{n=2}^{\infty}\frac{a_n}{\log(n)} converges. I have tried...
  15. Aristotle

    Question about finding min. sum of product using K-maps?

    Homework Statement Figure out the minimum sum of products for g(r s t) = r't' + rs' +rs 2. The attempt at a solution I understand you can simplify it with the Boolean theorems (e.g r't' + r = t' + r) , however how would you solve it using K-maps? I drew out a truth table, but it seems as if...
  16. anemone

    MHB Evaluate Sum: $x^4/(x-y)(x-z)+y^4/(y-z)(y-x)+z^4/(z-x)(z-y)$

    Let $x=\sqrt{7}+\sqrt{5}-\sqrt{3},\,y=\sqrt{7}-\sqrt{5}+\sqrt{3},\,z=-\sqrt{7}+\sqrt{5}+\sqrt{3}$. Evaluate $\dfrac{x^4}{(x-y)(x-z)}+\dfrac{y^4}{(y-z)(y-x)}+\dfrac{z^4}{(z-x)(z-y)}$.
  17. RJLiberator

    Sum of Unitary Matrices Question

    Homework Statement Find an example of two unitary matrices that when summed together are not unitary. Homework EquationsThe Attempt at a Solution A = \begin{pmatrix} 0 & -i\\ i & 0\\ \end{pmatrix} B = \begin{pmatrix} 0 & 1\\ 1 & 0\\ \end{pmatrix} A+B = A = \begin{pmatrix} 0 & 1-i\\ 1+i &...
  18. RJLiberator

    Sum of Hermitian Matrices Proof

    Homework Statement Show that the sum of two nxn Hermitian matrices is Hermitian.Homework Equations Hermitian conjugate means that you take the complex conjugate of the elements and transpose the matrix. I will denote it with a †. I will denote the complex conjugate with a *. The Attempt at a...
  19. Greg

    MHB Trig proof: sum of squared cosecants

    Hi! I've tried a couple of approaches with this: converting to complex exponential form and using standard trigonometric identities but have been unable to solve. I suspect DeMoivre's formula applies but I don't see how.Prove...
  20. P

    Convergence of Divergent Series

    Consider the two divergent series: $$\sum_{n=k}^{\infty} a_n$$ $$\sum_{n=k}^{\infty} b_n$$ Is it possible for ##\sum_{n=k}^{\infty} (a_n \pm b_n)## to converge?
  21. K

    Why are the normal forces for the legs pointing downwards?

    Homework Statement Hi all! I have been blindly drawing normal forces till today and i stumbled on this question. Homework EquationsThe Attempt at a Solution I have drawn 3 normal forces, - Hand - Leg - Center of mass Is there a normal force at the center of mass and is it the sum of the...
  22. S

    What is the frequency of the sum of several sine waves?

    I am given three sine waves with individual frequency being 10 Hz, 50 Hz, and 100 Hz. What is the frequency of the following : y(t) = sin(2π10t) + sin(2π50t) + sin(2π100t) Is it simply 100, the LCM of all the sin waves? If not, How to calculate the frequency of y(t) ?
  23. Illmatic1

    MHB Reciprocals of Roots in a Polynomial Equation

    Let $a, b, c, d, e$, and $f$ be the roots of $x^6 + 15x^5 + 53x^4 -127x^3 -1038x^2 -1832x- 960 = 0.$ Find $\displaystyle \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f}.$
  24. Illmatic1

    MHB What is the formula to find the sum of $\cot^2$ for a given range?

    Find the value of $ \displaystyle \sum_{k=1}^{n} \cot^2\bigg(\frac{\pi k}{2n+1} \bigg) $
  25. B

    Finding the Sum of A + \sqrt{A^2 - B^2} and U + \sqrt{U^2 - V^2}

    Given two numbers: A + \sqrt{A^2 - B^2} and U + \sqrt{U^2 - V^2} OBS: A, B, U and V are real numbers. I want sum it and express the result in the same form: A + \sqrt{A^2 - B^2} + U + \sqrt{U^2 - V^2} = x + \sqrt{x^2 - y^2} So, x depends of A and U. And y depends of B and V: x = x(A, U)...
  26. Albert1

    MHB Max n for Sum of 3 Numbers Multiple of 27 in A

    $A=\begin{Bmatrix} {1,2,3,4,5,------,2015} \end{Bmatrix}$ if we pick $n$ numbers from $A$, we call it the set $B$ ,and the sum of any three numbers from $B$ are multiple of 27 ,find $max(n)$ , and the largest number we can choose from $A$
  27. B

    The sum of elastic and gravitational energy

    Homework Statement 1. What is the gravitational energy (relative to the unstretched surface of the trampoline) of the 20kg ball at its apex 2.0m above the trampoline 2. What is the kinetic energy of the ball just before impacting the trampoline 3. At maximum stretch at the bottom of the motion...
  28. Albert1

    MHB Roots of Equations & Sum of Inverses: $a=1,2,3,\dots,2011$

    $a=1,2,3,4,5,------2011$, the roots of the equations $x^2-2x-a^2-a=0,$ are : $(\alpha_1,\beta_1),(\alpha_2,\beta_2),----------,(\alpha_{2011},\beta_{2011})$ respectively please find : $\sum_{n=1}^{2011}(\dfrac{1}{\alpha_n}+\dfrac {1}{\beta_n})$
  29. anemone

    MHB What is the Sum of Positive Integers a, b, and c Given a Specific Equation?

    If $a,\,b$ and $c$ are positive integers such that $16a b c+4a b+4a c+4b c+ a+b+c =4561$, find the sum of $a+b+c$.
  30. F

    Geometry Problem - Sum of distances

    Homework Statement ABC is a triangle with I as the centre of the incircle and K as the centre of the circumcircle (I ≠ K). d(X) is the sum of the distances from a point X inside the triangle to the three sides. Verify if d(P)=d(Q) (P and Q are points inside the triangle), the lines PQ and IK...
  31. A

    MHB History of Sum of Squares: Pythagoras & Beyond

    I would like to know some history on the subject like who is the first to think about sum of squares of integers and what he/she was thinking about. I think maybe it is related to Pythagorean triples. Thanks
  32. S

    MHB Proving the Limit of an Infinite Sum

    Prove that$\lim_{{n}\to{\infty}}\sum_{j=0}^{n} {n \choose j} \frac{{(x-a)}^{n+j}}{(n+j) !} = 0 $ thanks Sarrah
  33. D

    Sum of Related Periodic Functions

    I have been looking through the book Counterexamples: From Elementary Calculus to the Beginning of Calculus and became interested in the section on periodic functions. I thought of the following question: Suppose you have a periodic real valued function f(x) with a fundamental period T. Let c...
  34. D

    Sum of Two Periodic Orthogonal Functions

    Homework Statement This problem is not from a textbook, it is something I have been thinking about after watching some lectures on Fourier series, the Fourier transform, and the Laplace transform. Suppose you have a real valued periodic function f with fundamental period R and a real valued...
  35. Saitama

    MHB Writing a number as sum of squares

    Here's the problem statement from HackerRank: https://www.hackerrank.com/contests/programaniacs-june-15/challenges/sum-of-squares-1 Since the constraints are small, I tried a DP solution. Code I have written so far: #include <cmath> #include <cstdio> #include <vector> #include <iostream>...
  36. P

    Prove Sum Approximation Theorem

    Homework Statement I put up the image so that you can see the hints if you're curious. I am supposed to prove that if ## S=\sum_{n=0}^{\infty}a_{n}x^{n}## converges for ##|x|<1##, and if ##|a_{n+1}|<|a_{n}|## for ##n>N##, then $$|S-\sum_{n=0}^{N}a_{n}x^{n}|<|a_{N+1}x^{N+1}|\div (1-|x|)$$...
  37. A

    MHB Sum of infinite divergent series

    It is well known that the below series are divergent $1 - 1 + 1 - 1 + \cdots $ $1 - 2 + 3 - 4 + \cdots $ $1 + 2 + 3 + \cdots $ But after i watched a video in youtube for the channel " Numberphile " they proved that the first is equal to 1/2 , 1/4 and the last one is -1/12 ! The way to...
  38. Keen94

    Sum of Consecutive Powers

    Homework Statement Use the method of Problem 6 to show that ∑1≤k≤n kp can always be written in the form (np+1) / (p+1) +Anp+Bnp-1+Cnp-2+... Source: Calculus by Michael Spivak. Chapter 2 problem 7. Homework Equations The method from problem 6 is described as follows: The formula for the sum of...
  39. matqkks

    Sum of Two Squares: Intro to Number Theory

    Why bother writing a given integer as the sum of two squares? Does this have any practical application? Is there an introduction on a first year number theory course which would motivate students to study the conversion of a given integer to sums of two squares?
  40. matqkks

    MHB Sum of Two Squares: Applications & Motivation

    Why bother writing a given integer as the sum of two squares? Does this have any practical application? Is there an introduction on a first year number theory course which would motivate students to study the conversion of a given integer to sums of two squares?
  41. Rectifier

    Geometric sum - Alfred & interest-rate

    Homework Statement Alfred puts 985 USD on his bank account every time he has a birthday. Alfred just turned 48. He started to save money when he turned 35 (including 35th birthday). How much money is there on his savings-account if the interest-rate was 3.7% every year and that he had no money...
  42. patrickbotros

    Parameterization of Sum of Squares

    I've seen the parameterization of a^2+b^2=c^2 and also a^2+b^2=c^2+d^2, but I don't know how they arrived at those parameterizations. Would it be possible to parameterize something with two equalities like a^2+b^2=c^2+d^2=e^2+f^2? Any help is appreciated!
  43. W

    What is the coinciding point limit of these two parametrized terms?

    Homework Statement The story is that I would like to evaluate the coinciding point limit (when ## (x^0, x^1)→(y^0,y^1)##) of these two terms: \begin{eqnarray*} &&\frac{1}{2L}e^{\frac{i}{2}eE\left((x ^1)^2-(y^1)^2\right)}\left( im\left( x^0-y^0+ x^1-y^1\right) \right)...
  44. Hanyu Ye

    Sum formula for the modified Bessel function

    Hi, everybody. Mathematic handbooks have given a sum formula for the modified Bessel function of the second kind as follows I have tried to evaluate this formula. When z is a real number, it gives a result identical to that computed by the 'besselk ' function in MATLAB. However, when z is a...
  45. Saitama

    C/C++ How can I optimize my C++ code for a sum problem on CodeChef?

    I am trying this problem on CodeChef: Just a simple sum My task is to evaluate: $$\sum_{i=1}^n i^i \pmod m$$ Following is the code I have written: #include <iostream> using namespace std; typedef long long ll; ll modularPower(ll base, ll exponent, ll M) { ll res = 1; while...
  46. A

    MHB What is the sum of polynomial zeros?

    From Vieta's Formulas, I got: $a=2r+k$ $b=2rk+r^2+s^2$ $65=k(r^2+s^2)$ Where $k$ is the other real zero. Then I split it into several cases: $r^2 + s^2 = 1, 5, 13, 65$ then: For case 1: $r = \{2, -2, 1, -1 \}$ $\sum a = 2(\sum r) + k \implies a = 13$ Then for case 2: $r^2 + s^2 = 13$, it...
  47. P

    Write the Maclaurin series for (1+x)^(-1/2) as a sum

    Homework Statement Write the Maclaurin series for ##\frac{1}{(1+x)^{1/2}} ## in ##\sum## form using the binomial coefficient notation. Then find a formula for the binomial coefficients in terms of n. Homework Equations 3. The Attempt at a Solution [/B]...
  48. brainbaby

    Making the sum of 2 resistors independent of 1 of them

    I am really not sure about this ... I may be generalizing it ...but anyway... suppose we have two resistors in series...connected by ofcourse a voltage source...and a certain amount of total current is flowing through the whole circuit.. now if we want to make the total current totally...
  49. nuuskur

    Interval of convergence and sum of power series

    Homework Statement Consider ##\sum\limits_{n=0}^{\infty} \frac{n+1}{(2n)!}(x+1)^{2n+1}##. Find the interval of convergence and sum of the power series. Homework EquationsThe Attempt at a Solution According to the textbook: given the power series ##\sum a_n(x-c)^n## the radius of convergence...
  50. jk22

    Is another definition of sum useful?

    Von neumann and bell pointed out that basically the non isomorphic fact that the spectrum $$\sigma(A+B)!=\sigma(A)+\sigma(B)$$ leads to contradictions. If we but replace the sum by $$A\otimes 1+1\otimes B$$ then the above inequality becomes an equality. This would make things much easier. We...
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