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. F

    Find the standard Sum Of Products and Product Of Sums forms of the solution

    Homework Statement A switching network has 4 inputs and a single output (Z) as shown in the figure below. The output Z is 1 iff the binary number represented by ABCD ( A is the MSB) is an even number greater than 5. Find : a) The standard POS of Z (abbreviated form). b) The standard SOP of...
  2. M

    Delay and Sum Beamforming Equation Derivation

    Homework Statement I have to simplify this beam form (equation 1) which simplifies to equation 2 and then finally to equation 3. Homework Equations equation 1: e^-ix((1-e^y)/(1-e^z)) where x = Beta*M_(1/2), y = beta*M, z= Beta equation 2: sin(M*Beta/2)/(sin(Beta/2)) equation 3...
  3. K

    Using the Integral Test to Show Sum is Less Than pi/2

    Homework Statement Use the integral test to show that the sum of the series gif.latex ##\sum_{n=1}^\infty \dfrac{1}{1+n^2}## is smaller than pi/2. Homework EquationsThe Attempt at a Solution I know that the series converges, and the integral converges to pi/4. As far as I´ve understood...
  4. J

    MHB Sum of two infinite series: Σ[1/(36r^2-1)+2/(36r^2-1)^2]

    Evaluation of \displaystyle \sum_{r=1}^\infty \left(\frac{1}{36r^2-1}+\frac{2}{(36r^2-1)^2}\right)
  5. J

    Using binomial coefficients to find sum of roots

    Homework Statement >Find the sum of the roots, real and non-real, of the equation x^{2001}+\left(\frac 12-x\right)^{2001}=0, given that there are no multiple roots. While trying to solve the above problem (AIME 2001, Problem 3), I came across three solutions on...
  6. Mr Davis 97

    I Showing that an infinite sum converges when a sign in the denominator is flipped

    ##\displaystyle \sum_{n=1}^\infty\frac{1}{n^2+n/2}## converges by the direct comparison test: ##\displaystyle \left|\frac{1}{n^2+n/2}\right| \le \left|\frac{1}{n^2}\right|##, and ##\displaystyle \sum_{n=1}^\infty\frac{1}{n^2} = \frac{\pi^2}{6}##. But what if we want to show that ##\displaystyle...
  7. lfdahl

    MHB Infinite sum: Evaluate (2x-1)/(1-x+x^2)+(4x^3-2x)/(1-x^2+x^4)+(8x^7-4x^3)/(1-x^4+x^8)

    For 0 < x< 1, find the sum: \[\frac{2x-1}{1-x+x^2}+\frac{4x^3-2x}{1-x^2+x^4}+\frac{8x^7-4x^3}{1-x^4+x^8}+ ...\]
  8. MrsTesla

    Variance with Poisson distribution

    <Moderator's note: Moved from a technical forum and thus no template.> So, I have this problem and I am stuck on a sum. The problem I was given is the following: The probability of a given number n of events (0 ≤ n < ∞) in a counting experiment per time (e.g. radioactive decay events per...
  9. Hafsaton

    Prove: Sq Root of a Sum ≤ Sum of the Sq Roots

    Homework Statement (x.y)ER+ that means x and y >=0 Homework Equations Prove that n√(x+y)<=n√x + n√y The Attempt at a Solution
  10. B

    B Force Resultant is equal to the sum of the components -- why?

    Why are the resultant's X and Y components of two forces equal the sum of the X and Y components of the two forces?
  11. I

    Can Direct Sums and Projections Fully Describe Subspaces in Linear Algebra?

    Homework Statement Let ##V = \mathbb{R}^4##. Consider the following subspaces: ##V_1 = \{(x,y,z,t)\ : x = y = z\}, V_2=[(2,1,1,1)], V_3 =[(2,2,1,1)]## And let ##V = M_n(\mathbb{k})##. Consider the following subspaces: ##V_1 = \{(a_{ij}) \in V : a_{ij} = 0,\forall i < j\}## ##V_2 =...
  12. Mr Davis 97

    Proving "Limits of Finite Sequences Implies Limit of Sum

    Homework Statement For each ##n\in\mathbb{N}##, let the finite sequence ##\{b_{n,m}\}_{m=1}^n\subset(0,\infty)## be given. Assume, for each ##n\in\mathbb{N}##, that ##b_{n,1}+b_{n,2}+\cdots+b_{n,n}=1##. Show that ##\lim_{n\to\infty}( b_{n,1}\cdot a_1+b_{n,2}\cdot a_2+\cdots+b_{n,n}\cdot a_n) =...
  13. S

    I QED sum over polarization states

    Hello! In the calculation of the QED matrix element, it says in the book I read that we have to sum over the polarization states of the photon: $$\sum_\lambda \epsilon_\mu^\lambda\epsilon_\nu^{\lambda *}=-g_{\mu\nu}$$ I am a bit confused why do we do a summation over the orthonormal basis...
  14. bhobba

    A Ramanujan Summation and ways to sum ordinarily divergent series

    Hi All Been investigating lately ways to sum ordinarily divergent series. Looked into Cesaro and Abel summation, but since if a series is Abel Mable it is also Cesaro sumable, but no, conversely,haven't worried about Cesaro Summation. Noticed Abel summation is really a regularization...
  15. Urmi

    Gravitation sum related to Centripetal Acceleration

    Homework Statement The distance between the centres of the Earth and the moon is 60 times the radius of the earth. Calculate the centripetal acceleration of the moon. Acceleration due to gravity on the Earth's surface is 10m/s. Homework Equations Centripetal acceleration= v^2/R Orbital...
  16. opus

    B Partial Fraction Decomposition - "Telescoping sum"

    There is a problem in a PreCalculus book that I'm going over that states: Express the sum ##\frac{1}{2⋅3}+\frac{1}{3⋅4}+\frac{1}{4⋅5}+...+\frac{1}{2019⋅2020}## as a fraction of whole numbers in lowest terms. It goes on to state that each term in the sum is of the form...
  17. CharlieCW

    Finding Eigenvalues of an Operator with Infinite Basis

    I just began graduate school and was struggling a bit with some basic notions, so if you could give me some suggestions or point me in the right direction, I would really appreciate it. 1. Homework Statement Given an infinite base of orthonormal states in the Hilbert space...
  18. Pushoam

    Dimensionality of the sum of subspaces

    Homework Statement Suppose that ## \mathbb {V}_1^{n_1} ## and ## \mathbb {V}_2^{n_2} ## are two subspaces such that any element of ## \mathbb {V}_1^{n_1} ## is orthogonal to any element of ## \mathbb {V}_2^{n_2} ## . Show that dimensionality of ## \mathbb {V}_1^{n_1} + \mathbb {V}_2^{n_2}...
  19. J

    MHB Find the total sum of money shared by the three girls.

    Kate, Nora, and Devi shared a sum of money. Kate received 24 dollars and Nora received x dollars more than Kate. Devi received 2x dollars more than kate a) Find the sum of money shared in terms of x. my answer: total = 24 + (x+24) + 2(24)b) Nora received $30. Find the total sum of money shared...
  20. M

    MHB How Can Riemann Sums Calculate the Volume of a Tepee Tent?

    THE QUESTION By using Riemann’s sum, synthesise a mathematical model for finding the exact volume of any ‘tepee’ tent of side s and height h. HERE'S WHAT I HAVE Am currently stuck on writing a side length for the hexagon at any height 'x'
  21. anemone

    MHB Find the sum of a^(1/3)+b^(1/3)+c^(1/3)

    Let $a,\,b$ and $c$ be real numbers such that $a+b+c=ab+bc+ac=-\dfrac{1}{2}\\abc=\dfrac{1}{8}$ Evaluate $a^{\tiny\dfrac{1}{3}}+b^{\tiny\dfrac{1}{3}}+c^{\tiny\dfrac{1}{3}}$.
  22. M

    MHB Sum and product of roots of quadratic equation

    #85
  23. M

    MHB Riemanns Sum Problem: Find the exact volume

    This is what i have so far We can find the exact volume of any shape using: V= int[a,b] A(x) dx Where,A(x)is the cross-sectional area at height x and [a,b] is the height interval We know that the horizontal cross-sections are hexagonal ∴A=(3√3)/2 a^2 Where a,is the length of a side Write the...
  24. H

    MHB Vatha & Chris Ages: Solve to Find Out How Old They Are Now

    Q: The sum of the present ages of Vatha and Chris is 36. In 4 years time, the sum of their ages will equal twice Vatha's present age. How old are they now?A: Vatha:22, Chris: 14 (from the back of the textbook, only I'm not sure how to get here)I'm a little stuck with this one. If you could...
  25. Philip Koeck

    A Why is the partition for Fermions a sum of Boltzman factors?

    The partition function should essentially be the sum of probabilities of being in various states, I believe. Why is it then the sum of Boltzmann factors even for fermions and bosons? I've never seen a good motivation for this in literature.
  26. L

    A Convergence of a subsequence of a sum of iid r.v.s

    ##X_i## is an independent and identically distributed random variable drawn from a non-negative discrete distribution with known mean ##0 < \mu < 1## and finite variance. No probability is assigned to ##\infty##. Now, given ##1<M##, a sequence ##\{X_i\}## for ##i\in1...n## is said to meet...
  27. binbagsss

    Characteristic function of the sum of random variables

    Homework Statement I am trying to understand the very last equality for (let me replace the tilda with a hat ) ##\hat{P_{X}(K)}=\hat{P(k_1=k_2=...=k_{N}=k)}##(1) Homework Equations I also thought that the following imaginary exponential delta identity may be useful, due to the equality of...
  28. Math Amateur

    MHB Finite Sum of Indecomposable Modules .... Bland, Proposition 4.2.10 .... ....

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.10 ... ... Proposition 4.2.10 reads as follows:My questions are as follows:Question 1 In the...
  29. Mr Davis 97

    I Analyzing the Convergence of a Geometric Series

    I have the following series that I came up with in doing a problem: ##\displaystyle \sum_{n=0}^{\infty} \frac{1}{2^{n+1}(n+1)}##. I looked at WolframAlpha and it says that this series converges to ##\log (2)##. Is it possible to figure this out analytically?
  30. Arman777

    I Solve Coin Sum Problem: Find Combinations for 200p

    Let's suppose there are 4 types of coin, 1p 2p 5p and 10p. The problem is, we need to find total combinations which the sum of the coins gives us the 200p. I am trying to find a mathematical equation to solve the problem but I am stuck. First I started to think algebraically. And I have this...
  31. I

    [Linear Algebra] Sum & Direct Sum of Subspaces

    ⇒Homework Statement [/B] Calculate ##S + T## and determine if the sum is direct for the following subspaces of ##\mathbf R^3## a) ## S = \{(x,y,z) \in \mathbf R^3 : x =z\}## ## T = \{(x,y,z) \in R^3 : z = 0\}## b) ## S = \{(x,y,z) \in \mathbf R^3 : x = y\}## ## T = \{(x,y,z) \in \mathbf R^3 ...
  32. W

    Java Java: Which Term determines this is a Sum?

    Hi all, I have the code below with a 'for' loop. I see the output is 17, the sum of all elements of the (variable?) result. I am a bit confused: I cannot tell which part of the code is used to determine that the values {1,2,10,4,5} must be added to each other? Please comment/correct: 1)We define...
  33. S

    Boolean Algebra, Minimum Sum of Products Problem

    Hello to everyone who's reading this. The problem I need help with is the following.: Homework Statement "Simplify to obtain minimum SOP. F(A, B, C, D) = A’B’CD’+AC’D’+ABC’+AB’C+AB’C+BC’D" The problem stated above has two provided solutions, the "main" one and the "alternate" one. I'm...
  34. binbagsss

    Elliptic Functions Proof of Sum of Residues=0

    Homework Statement Hi I am looking at the attached proof for this property. I agree with the first line due to periodicity, but unsure about the next- see below 3)attempt Homework Equations To me, I deemed the integration substituion rule as relevant to this question, but perhaps...
  35. Math Amateur

    I External Direct Sums and the Sum of a Family of Mappings ....

    I have an issue/problem that relates to Bland initial treatment of external direct sums including Proposition 2.1.5 ... especially Bland's definition of the sum of a family of mappings ... Bland's text on this is as follows: In the above text by Bland we read the following: " ... ... We now...
  36. S

    A On a finitely generated submodule of a direct sum of modules....

    I am new on this forum, this is my gift for you. Suppose ##(M_i)_{i \in I}## is a family of left ##R##-modules and ##M = \bigoplus_{i \in I} M_i## (external direct sum). Suppose ##N = \langle x_1, \cdots ,x_m \rangle## is a finitely generated submodule of ##M##. Then for each ##j = 1, \cdots...
  37. S

    Sum of (n+1) terms in exponential series

    Homework Statement S = 1+ x/1! +x2/2! +x3/3! +...+xn/n! To find S in simple terms. Homework Equations None The Attempt at a Solution I tried with Taylor's expansion, coshx and sinhx expansions. But cannot see consequence.
  38. J

    A Sum of independent random variables and Normalization

    Hi, Lets say I have N independent, not necessarily identical, random variable. I define a new random variable as $$Y=Σ^{N}_{i=0} X_{i}$$ does Y follow a normalized probability distribution?
  39. Luck0

    I Inverse of the sum of two matrices

    Suppose I have a matrix M = A + εB, where ε << 1. If A is invertible, under some assumptions I can write e Neumann series M-1 = (I - εA-1B)A-1 But if A is not invertible, how can I expand M-1 in powers of ε? Thanks in advance
  40. MarkFL

    MHB Can the Partial Sum of a Difficult Series be Solved with Induction?

    Hello MHB! (Wave) A young man in high school I know has been essentially tasked with finding the following partial sum: S=\sum_{k=0}^{n}\left(\frac{2^k}{3^{2^k}+1}\right) I honestly have no idea how to proceed, and I am hoping someone here can provide some insight. (Star)
  41. M

    MHB How Many Rolls of Dice to Get Sum 12?

    How many times do we need to throw 2 dices to get more than 1/2 probability that at least once the sum of the two dices will be 12.
  42. Amlan mihir

    Find the value of the trigonometric sum

    Homework Statement i have no idea how to proceed
  43. S

    MHB On a finiteley generated submodule of a direct sum of left R-modules

    Suppose $(M_i)_{i \in I}$ is a family of left $R$-modules and $M = \bigoplus_{i \in I} M_i$. Suppose $N = \langle x_1 \cdots x_m \rangle$ is a finitely generated submodule of $M$. Then for each $j = 1 \cdots m$, there is a finite $I_j \subset I$ such that $x_j \in \bigoplus_{i \in I_j} M_i$...
  44. EEristavi

    Discovering the Formula for Σ (i=1, n) √i

    Homework Statement Calculate Σ (i=1, n) √i I want to write general formula, then use it for any n (like we have for Σ (i=1, n) i Homework Equations Σ (i=1, n) i = n (n+1) / 2 Σ (i=1, n) i^2 = n (n+1)(2n +1) / 6The Attempt at a Solution Comparing formulas provided above: I assume the answer...
  45. Physics345

    Finding the least possible sum of a product.

    Homework Statement The product of two positive numbers is 100. What numbers will produce the least possible sum? Confirm that the sum is in fact a minimum. Homework EquationsThe Attempt at a Solution For this question here I feel like the wording is a bit confusing, I tried my best please let...
  46. C

    MHB The proof of the infinite geometric sum

    Dear Everybody, I need some help with find M in the definition of the convergence for infinite series. The question ask, Prove that for $-1<r<1$, we have $\sum_{n=0}^{\infty} r^n=\frac{1}{1-r}$. Work: Let $\sum_{n=0}^{k} r^n=S_k$. Let $\varepsilon>0$, we must an $M\in\Bbb{N}$ such that $k\ge...
  47. F

    Calculate the sum of all high-order bytes in array NUM1

    Homework Statement Calculate the sum of all hight-order bytes in array NUM1 and store the sum in a memory location named newH. Define newH as needed. Homework Equations - The Attempt at a Solution INCLUDE Irvine32.inc .data NUM1 sword 1h,2h,3h,4h,5h,6h,7h,8h,9h,10h,11h,12h,13h,14h,15h,16h...
  48. howabout1337

    B How can I show the sum results in this?

    ##\sum_{n=0}^\infty \frac 1{p^{nz}}=1+\frac1{p^z}+\frac1{p^{2z}}+\frac1{p^{3z}}...## ##\frac 1{p^z}\sum_{n=0}^\infty \frac 1{p^{nz}}=\frac1{p^z}+\frac1{p^{2z}}+\frac1{p^{3z}}+\frac1{p^{4z}}...## something happens and it shows: ##\frac 1{p^z}\sum_{n=0}^\infty \frac...
  49. MountEvariste

    MHB Sum of powers limit via Riemann sums?

    One of the many excellent problems by lfdahl in the challenge questions and puzzles subforum was recently: https://mathhelpboards.com/challenge-questions-puzzles-28/prove-limit-23480.html My first idea was Riemann sums! I didn't succeed. So I ask, can this limit be calculated via Riemann...
  50. lfdahl

    MHB Find the infinite sum of fractions 2/(3⋅5)+(2⋅4)/(3⋅5⋅7)+(2⋅4⋅6)/(3⋅5⋅7⋅9)+....

    I´m not sure, whether this little challenge has been posted before. I have searched the forum and didn´t find it. It might still be a duplicate though ... Find the sum of fractions $$\frac{2}{3\cdot5}+\frac{2\cdot4}{3\cdot5\cdot7}+\frac{2\cdot4\cdot6}{3\cdot5\cdot7\cdot9}+...$$
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