# 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. ### A little help understanding 3 phase waveforms and summation

Hello all, I'm in the process of learning about 3 phase power and how to wire loads to a generator. I've searched high and low with as many sentence structures as I could think of in the google search bar and I can't seem to find the answers I'm looking for. I'm really hoping you guys can...
2. ### POTW Sum of an Alternating Series

Find, with proof, the sum of the alternating series $$\sum_{n = 0}^\infty \frac{(-1)^n}{(2n+1)^3}$$
3. ### Use method of difference to find sum of series

My interest is on the (highlighted part in yellow ) of finding the partial fractions- Phew took me time to figure out this out :cool: My approach on the highlighted part; i let ##(kr+1) =x ## then, ##\dfrac{1}{(kr+1)(kr-k+1)} = \dfrac{1}{x(x-k)}## then...
4. ### Find sum of all odd square numbers

There is a mistake on this textbook (The mistake is pretty obvious) but hey I hope I did not miss something... ought to be ##40425 - 19600##
5. ### Find the sum of the coefficients in the expansion ##(1+x)^n##

##(1+x)^n=1+C_1x+C_2x^2+C_3x^3...+C_nx^n## Let ##x=1##, hence ##2^n=1+C_1+C_2+C_3...+C_n## which is equal to the sum of the coefficients. I originally thought the sum of the coefficients would be ##2^n-1## since the very first term ##1## is just a number and has no variable. But apparently...
6. ### Finding a definite integral from the Riemann sum

Hi! I am having trouble finalizing this problem. The interval is given so we know that a = 1 and b = 2. From there you can figure out that ∆x = 1/n, xiR = 1 + i/n. Using logarithmic properties, I rearranged the expression and wrote (1 + i/n)(1/n)ln[(n + i)/n]. I can guess that the function is...
7. ### Identifying variables from Riemann sum limits

Hi! I understand that this is an expanded Riemann sum but I'm having trouble determining its original form. I don't actually have any ideas as to how to find it, but I know that once I determine the original form of the Riemann sum, I will be able to figure out the values for a, b, and f. If...
8. ### A formula involving the sum of cosines of the angles of a triangle

Problem Statement : The statement appeared on a website where a different problem was being solved. I got stuck at the (first) statement in the solution that I posted above 👆. Here I copy and paste that statement from the website, which I cannot show : Attempt : To save time typing, I write...
9. ### I Upper and Lower Darboux Sum Inequality

Lemma Let f be a bounded function on [a,b]. If P & Q are partitions of [a,b] and P ##\subseteq## Q , then L(f,P) ##\leq## L(f,Q) ##\leq## U(f,Q) ##\leq## U(f,P) . Question is "How can P have bigger upper darboux sum than Q while it is a subset of Q"
10. ### Sum of angles with x,y and z axis made by a vector

I want to know if there is any proper relation between the angles of a vector with the three dimensional coordinate axes, if the angles are ,α , β and γ, will the sum of α, β and γ be 180 degress that is α + β + γ = 180°,m finding the same to be true in a 2 D case where α + β = 90° and γ =...
11. ### I Understanding tensor product and direct sum

Hi, I'm struggling with understanding the idea of tensor product and direct sum beyond the very basics. I know that direct sum of 2 vectors basically stacks one on top of another - I don't understand more than this . For tensor product I know that for a product of 2 matrices A and B the tensor...
12. ### Solve the problem involving sum of a series

Attempt; ##\dfrac{1}{r(r+1)(r+2)} -\dfrac{1}{(r+1)(r+2)(r+3)}=\dfrac{(r+3)-1(r)}{r(r+1)(r+2)(r+3)}=\dfrac{3}{r(r+1)(r+2)(r+3)}## Let ##f(r)=\dfrac{1}{r(r+1)(r+2)}## ##f(r+1)= \dfrac{1}{(r+1)(r+2)(r+3)}## Therefore ##\dfrac{3}{r(r+1)(r+2)(r+3)}## is of the form ##f(r)-f(r+1)## When...
13. ### Solve the problem involving sum of a series

My attempt; ##r^2+r-r^2+r=2r## Let ##f(r)=(r-1)r## then it follows that ##f(r+1)=r(r+1)## so that ##2r## is of the form ##f(r+1)-f(r)##. When ##r=1;## ##[2×1]=2-0## ##r=2;## ##[2×2]=6-2## ##r=3;## ##[2×3]=12-6## ##r=4;## ##[2×4]=20-12## ... ##r=n-1##, We shall have...
14. ### A How to sum an infinite convergent series that has a term from the end

From my physical problem, I ended up having a sum that looks like the following. S_N(\omega) = \sum_{q = 1}^{N-1} \left(1 - \frac{q}{N}\right) \exp{\left(-\frac{q^2\sigma^2}{2}\right)} \cos{\left(\left(\mu - \omega\right)q\right)} I want to know what is the sum when N \to \infty. Here...
15. ### I Randomly Stopped Sums vs the sum of I.I.D. Random Variables

I've came across the two following theorems in my studies of Probability Generating Functions: Theorem 1: Suppose ##X_1, ... , X_n## are independent random variables, and let ##Y = X_1 + ... + X_n##. Then, ##G_Y(s) = \prod_{i=1}^n G_{X_i}(s)## Theorem 2: Let ##X_1, X_2, ...## be a sequence of...
16. ### I Feynman diagrams and sum over paths

Hi ! In a Feynman diagram, can we consider that the propagator specifying the transition amplitude of a particle (let's say, of a "real" electron, or of a "virtual" photon) between two points or two vertices, is in fact itself the sum of a multiplicity of probability amplitudes, each one...
17. ### What is the formula for finding the nth partial sum?

Since we are adding numbers produced according to a fixed pattern, there must also be a pattern (or formula) for finding the sum. Hi, We use this method to find the ##S_n##. I don't understand how the sum will also be in a pattern. Can someone please explain this line in bold?
18. ### Express a function as a sum of even and odd functions

I am refreshing on this; of course i may need your insight where necessary...I intend to attempt the highlighted...this is a relatively new area to me... For part (a), We shall let ##f(x)=\dfrac{1}{x(2-x)}##, let ##g(x)## be the even function and ##h(x)## be the odd function. It follows...
19. ### POTW Find Triplets of Positive Integers with Sum of Cubes

Find all triples (a, b, c) of positive integers such that ##a^3+b^3+c^3=(abc)^2##.

21. ### Prove by induction the sum of complex numbers is complex number

See the work below: I feel like it that I did it correctly. I feel like I skip a step in my induction. Please point any errors.
22. ### POTW Find the Sum of Two Real Numbers | Solve a + b with Given Equations

If a and b are two real numbers such that ##a^3-3a^2+5a=1## and ##b^3-3b^2+5b=5##, evaluate a + b.
23. ### I Primes -- Probability that the sum of two random integers is Prime

im thinking i should just integrate (binominal distribution 1-2000 * prime probability function) and divide by integral of bin. distr. 1-2000. note that I am looking for a novel proof, not just some brute force calculation. (this isn't homework, I am just curious.)
24. ### B Neglected terms in integral sum

Hello. As is known, we can neglect high-order term in expression ##f(x+dx)-f(x)##. For ##y=x^2##: ##dy=2xdx+dx^2##, ##dy=2xdx##. I read that infinitesimals have property: ##dx+dx^2=dx## I tried to neglect high-order terms in integral sum (##dx^2## and ##4dx^2## and so on) and I obtained wrong...
25. ### MHB Closed form solution to sum of sine positive zero-crossings

Hello, I would like to know, if there's a closed form solution to the following problem: Given a sum of say, 3 sines, with the form y = sin(a.2.PI.t) + sin(b.2.PI.t) + sin(c.2.PI.t) where a,b,c are constants and PI = 3.141592654 and the periods in the expression are multiplication signs, what...
26. ### MHB I hope to sum up this series symbolically

Dear Colleagues I posted the same post in the group of Analysis. Perhaps it should have been posted here. It is a finite series for which I am seeking the sum. I tried using MATHEMATICA which did not work. Someone told me MAPLE will do it. So if one has it, I shall be thankful. All you have to...
27. ### B How to merge the sum and ##x^n##?

How do I merge ##x^n + \displaystyle\sum^x_{k=1} \frac{d^k}{d^kx} \frac{x^ny^k}{k}##? I tried changing the denominator of the summand to ##k+1## and make ##k## go from zero, but I had to divide by zero when k equaled one.
28. ### Numerical Analysis - Richardson Extrapolation on Riemann Sum

I got something like this, but I'm not sure it is correct, because if it has the same order of convergence as trapezoidal rule which is 2, it should yield the same result as trapezoidal rule but mine doesn't (?). For example sin(x) for [0,1], n with trapezoidal rule = 0.420735... With my own...
29. ### Find the sum of the series ##\sum_{r=n+1}^{2n} u_r##

Find question and solution here Part (i) is clear to me as they made use of, $$\sum_{r=n+1}^{2n} u_r=\sum_{r=1}^{2n} u_r-\sum_{r=1}^{n} u_r$$ to later give us the required working to solution... ... ##4n^2(4n+3)-n^2(2n+3)=16n^3+12n^2-2n^3-3n^2=14n^3+9n^2## as indicated. My question is on...
30. ### Solve the quadratic equation that involves sum and product

I am refreshing on this...Have to read broadly...i will start with (b) then i may be interested in alternative approach or any correction that may arise from my working. Cheers. Kindly note that i do not have the solutions to the following questions... For (b), we know that, say, if ##x=α##...
31. ### What is the remainder when the following sum is divided by 4?

Let ## n ## be an integer. Now we consider two cases. Case #1: Suppose ## n ## is even. Then ## n=2k ## for some ## k\in\mathbb{N} ##. Thus ## n^{5}=(2k)^{5}=32k^{5}\equiv 0 \pmod 4 ##. Case #2: Suppose ## n ## is odd. Then ## n=4k+1 ## or ## n=4k+3 ## for some ## k\in\mathbb{N} ##. Thus ##...
32. ### A Eigenvalue of the sum of two non-orthogonal (in general) ket-bras

We have a matrix ##M = \ket{\psi^{\perp}}\bra{\psi^{\perp}} + \ket{\varphi^{\perp}}\bra{\varphi^{\perp}}## The claim is that the eigenvalues of such a matrix are ##\lambda_{\pm}= 1\pm |\bra{\psi}\ket{\varphi}|## Can someone proof this claim? I have been told it is self-evident but I've been...
33. ### Every integer greater than 5 is the sum of three primes?

Proof: Let ## a>5 ## be an integer. Now we consider two cases. Case #1: Suppose ## a ## is even. Then ## a=2n ## for ## n\geq 3 ##. Note that ## a-2=2n-2=2(n-1) ##, so ## a-2 ## is even. Applying Goldbach's conjecture produces: ## 2n-2=p_{1}+p_{2} ## as a sum of two primes ## p_{1} ## and ##...
34. ### I What is wrong with crossing electric fields? Why can't you sum them?

I do not understand why electric fields cannot cross. Can't you just sum the two electric fields vectors to get a net electric field?
35. ### Show that the sum of twin primes ## p ## and ## p+2 ## is divisible?

Proof: Suppose ## p ## and ## p+2 ## are twin primes such that ## p>3 ##. Let ## p=2k+1 ## for some ## k\in\mathbb{N} ##. Then we have ## p+(p+2)=2k+1+(2k+1+2)=4k+4=4(k+1)=4m ##, where ## m ## is an integer. Thus, the sum of twin primes ## p ## and ## p+2 ## is divisible by ## 4 ##. Since ##...
36. ### I Distribution of Sum of Two Weird Random Variables....

Hi there. Let's say I have the following relationship: x = a + b*z + c*y z is distributed normally y is distributed according to a different distribution, say exponential Is there a way to figure out what is the distribution of x? Thanks!

38. ### I Probability of Sum of 2 Random Ints Being Prime

if I select two integers at random between 1 and 1,000, what is the probability that their sum will be prime?
39. ### Sum of Infinite Series: Finding the Value of S

Using the given rule for the ##x_n##, write $$\sum_l y_l = x_1 + \frac{1}{2} x_2 + \frac{1}{6} x_2 + \frac{1}{3} x_3 + \frac{1}{12} x_2 + \frac{1}{12} x_3 + \frac{1}{20} x_2 + \cdots + \frac{1}{n} x_n$$  = x_1 + \sum_{n=2}^\infty \frac{1}{n(n-1)} x_2 + \sum_{n=3}^\infty \frac{2}{n(n-1)} x_3...
40. ### I Sum of the dot product of complex vectors

Summary:: summation of the components of a complex vector Hi, In my textbook I have ##\widetilde{\vec{E_t}} = (\widetilde{\vec{E_i}} \cdot \hat{e_p}) \hat{e_p}## ##\widetilde{\vec{E_t}} = \sum_j( (\widetilde{\vec{E_{ij}}} \cdot {e_{p_j}}*) \hat{e_p}## For ##\hat{e_p} = \hat{x}##...
41. ### Each integer n>11 can be written as the sum of two composite numbers?

Proof: Suppose n is an integer such that ## n>11 ##. Then n is either even or odd. Now we consider these two cases separately. Case #1: Let n be an even integer. Then we have ## n=2k ## for some ## k\in\mathbb{Z} ##. Consider the integer ## n-6 ##. Note...

44. ### MHB Parity: Show that the sum contains at least one even number.

Hello, guys! I have a question that need help! A number with 17 digits is chosen and the order of its digits is inverted, forming a new number, These two numbers are then added up. Show that the sum contains at least one even number.
45. ### I Is the scalar magnetic Potential the sum of #V_{in}# and ##V_{out}##

Hi, I'm wondering if I have an expression for the scalar magnetic potential (V_in) and (V_out) inside and outside a magnetic cylinder and the potential is continue everywhere, which mean ##V^1 - V^2 = 0## at the boundary. Does it means that ##V^1 - V^2 = V_{in} - V_{out} = 0## ?
46. ### Solve the quadratic equation involving sum and product

For part (i), ##(x-α)(x-β)=x^2-(α+β)x+αβ## ##α+β = p## and ##αβ=-c## therefore,##α^3+β^3=(α+β)^3-3αβ(α+β)## =##p^3+3cp## =##p(p^2+3c)## For part (ii), We know that; ##tan^{-1} x+tan^{-1} y##=##tan^{-1}\left[\dfrac...
47. ### Solve the given quadratic equation that involves sum and product

For part a, We have ##α+β=b## and ##αβ =c##. It follows that, ##(α^2 + 1)(β^2+1)=α^2β^2+α^2+β^2+1)## =##α^2β^2+(α+β)^2-2αβ +1## =##c^2+b^2-2c+1## =##c^2-2c+1+b^2##...
48. ### I Absolute difference between increasing sum of squares

Given are non-negative integer variables ##x##, ##y## and ##z##. I am trying to deduce the absolute difference between a certain value of ##C=x^2+y^2+z^2## and the very next smallest increase in ##C## possible. I'd like to do this so I can (dis)prove the following: Whether small absolute...
49. ### Proving a sum of three squared terms, cyclic in #a,b,c#, is equal to 1

Problem statement : I copy and paste the statement of the problem from the text. (Given ##\boldsymbol{a+b+c=0}##) Attempt : I am afraid I couldn't make any meaningful progress. With ##a = -(b+c)##, I substituted for ##a## in the whole of the L.H.S, both numerators and denominators. I multiplied...
50. ### Multiple angles : Reducing the sum

Problem Statement : Let me copy and paste the problem as it appears in the text : Attempt : I haven't been able to make any significant attempt at solving this problem, am afraid. I tried to reduce all the higher submultiple angles ##2\theta, 4\theta, 8\theta## into ##\theta##, but the...