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

    MHB Finding sum of infinite geometric series

    find the sum of this infinite geometric series: 1 - √2 + 2 - 2√2 + ... a.) .414 b.) -2.414 c.) series diverges d.) 2 I found that the common difference is 2, so I calculated this: S∞= -.414/-1 s∞= .414 So i got that the answer is A, but will you check this?
  2. S

    B Uses for formulas for sum and product of quadratic roots

    Are there practical uses for the formulas for the sum and product of quadratic roots? I have only seen the topic for these sum and product formulas in one section of any college algebra and intermediate algebra books, and then nothing more. I'm just curious if people, ... scientists or...
  3. K

    Area between two graphs as a sum

    Homework Statement Homework Equations $$1^2+2^2+3^2+...+n^2=\frac{n(n+1)(2n+1)}{6}$$ The Attempt at a Solution $$S_n=2\{[f_1(x_1)-f_2(x_1)]\Delta x+[f_1(x_2)-f_2(x_2)]\Delta x+...+[f_1(x_{n-1})-f_2(x_{n-1})]\Delta x\}$$ $$S_n=2\{[18-\Delta x^2-\Delta x^2]+[18-(2\Delta x)^2-(2\Delta...
  4. K

    Why 1/2 is Coefficient in CK Sum for Integrals

    Homework Statement Why specifically 1/2 is the coefficient in CK? the sum, basically, doesn't change except for the coefficient. i can choose it as i want. I understand the sum must equal the integral but i guess that's not the reason Homework Equations Area under a curve as a sum...
  5. G

    Proper usage of Einstein sum notation

    Homework Statement I'm dealing with some pretty complex derivatives of a kernel function; long story short, there's a lot of summations going on, so I'm trying to write it down using the Einstein notation, for shortness and hopefully reduction of errors (also for the sake of a paper in which I...
  6. FallArk

    MHB How to calculate the lower and upper riemann sum

    I ran into some issues when trying to calculate the lower Riemann sum of f\left(x\right)={x}^{3}, x\in[0,1] I am asked to use the standard partition {P}_{n} of [0,1] with n equal subintervals and evaluate L(f,{P}_{n}) and U(f,{P}_{n}) What I did: L(f,{P}_{n}) =...
  7. Spinnor

    I Approximate a plane E&M wave with this large sum....

    I would like to approximate a plane electromagnetic wave with a very large sum of the following. Let an infinite line, say the z axis, have a electric polarization on that line and perpendicular to that line, say the x direction to be specific given by, P(z,t) = pcos(kz-ωt). The polarization...
  8. I

    MHB Sum of 2 Primes: 45 - (2 Digit Integer)?

    I think that we have to get all 2 digit odd numbers that can be expressed as the sum of 2 primes and subtract that from 45, so I think that the answer would be 45-(number of 2 digit integers n that are prime and have n-2 be prime as well)?
  9. Y

    I Finding the sum of heights under a curve

    In integral calc, you add up very small areas to find the total area under the curve. So it would be f(x1)Δx + f(x2)Δx+ ..., summed up. But what if you wanted to find out the sum of all heights under the curve? So it would be something like f(x1) + f(x2) + ... I'm thinking the formulation would...
  10. M

    MHB How do we find the sum of the roots in a quadratic equation?

    Show that the sum of the roots of the equation x^2 + px + q = 0 is -p. I need help with the set up. Is the discriminant involved here?
  11. E

    Why Isn't the Sum of Torques Zero in This Ladder Equilibrium Problem?

    Homework Statement Two ladders, 4.00 m and 3.00 m long, are hinged at point A and tied together by a horizontal rope 0.90 m above the floor (Fig. P11.89). The ladders weigh 480 N and 360 N, respectively, and the center of gravity of each is at its center. Assume that the floor is freshly waxed...
  12. Mr Davis 97

    Deriving the sum of sin and cos formula

    Homework Statement Show that ##a \sin x + b\cos x = c \sin (x + \theta)##, where ##c = \sqrt{a^2 + b^2}## and ## \displaystyle \theta = \arctan (\frac{b}{a})## Homework EquationsThe Attempt at a Solution We see that ##c \sin (x + \theta) = c \cos \theta (\sin x) + x \sin \theta (\cos x)##. So...
  13. karush

    MHB 10.02.10 Find the sum of the series

    $\textsf{Find the sum of the series}\\$ \begin{align*}\displaystyle S_{n}&=\sum_{n=1}^{\infty} \frac{4}{(4n-1)(4n+3)}=\color{red}{\frac{1}{3}} \\ \end{align*} $\textsf{expand rational expression } $ \begin{align*}\displaystyle \frac{4}{(4n-1)(4n+3)}...
  14. E

    Help in understanding sum of torque equation

    I've been reading my physics book and there they derived the formula ∑τ = Iα where τ is torque, I is moment of inertia of a rigid body and α is the angular acceleration. They did by taking an arbitrary particle on the rigid body with an applied external force tangent to the rotation. τ1 = Ftan *...
  15. terryds

    What is the value of the harmonic factorial series sum?

    Homework Statement What is the value of ## \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + ... ## ? Homework Equations [/B] I have no idea since it's neither a geometric nor arithmatic seriesThe Attempt at a Solution [/B] My Calculus purcell book tells me that it is e - 1 ≈...
  16. Isaac0427

    I Normalizing a Discrete Sum: Is the Wavefunction Fully Normalized?

    Say you have two energy eigenstates ##\phi_1## and ##\phi_2##, corresponding to energies ##E_1## and ##E_2##. The particle has a 50% chance of having each energy. The wavefunction would thus be ##\psi=\frac{\phi_1}{\sqrt{2}}+\frac{\phi_2}{\sqrt{2}}## Even though the coefficients are normalized...
  17. chakib

    B Sum of increasing and decreasing functions

    i want to know if any real function can be expressed as: f(x)=g(x)+h(x) such as g(x) is an increasing function and h(x) is a decreasing function? thanks
  18. D

    What is the PDF of the sum of n, iid, non central chi-square

    Homework Statement I need to find the pdf of sum of "n" iid non central chi-square distributed RV's. Homework Equations The PDF of the non-central chi-square RV is given herehttps://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution The Attempt at a Solution i tried to find the...
  19. M

    Infinite sum of non negative integers

    Homework Statement Consider a sequence of non negative integers x1,x2,x3,...xn which of the following cannot be true ? ##A)\sum ^{\infty }_{n=1} x_{n}= \infty \space and \space \sum ^{\infty }_{n=1} x_{n}^{2}= \infty## ##B)\sum ^{\infty }_{n=1} x_{n}= \infty \space and \space \sum ^{\infty...
  20. K

    B Applying L'Hospital's rule to Integration as the limit of a sum

    The definite integral of a function ##f(x)## from ##a## to ##b## as the limit of a sum is: $$\int_a^bf(x)dx=\lim_{h\rightarrow 0}h(f(a)+f(a+h)+.. ..+f(a+(n-2)h)+f(a+(n-1)h))$$ where ##h=\frac{b-a}{n}##. So, replacing ##h## with ##\frac{b-a}{n}## gives: $$\lim_{n\rightarrow...
  21. A

    Write 1729 as the sum of two cubes

    Homework Statement ##1729## can be written as ##12^3 + 1^3## and ##9^3 + 10^3## and ##7(10 + 9)(12 + 1)##. If ##x^3 + (7 - x)^3 = 1729##, use the above to find ##x##. ##x## is a non-integer Homework Equations ##1729 = 12^3 + 1^3 = 9^3 + 10^3 = 7(10 + 9)(12 + 1) = x^3 + (7 - x)^3## The Attempt...
  22. U

    MHB Sum of Infinite Series: Find 1/sqrt(2)

    Hey guys! I just have a question regarding finding the sum of an infinite series. Attached is the image of the question. I've tried to use the ratio test but it doesn't give me the result I need which happens to be 1/sqrt(2). I feel like this is one of those power series questions, but I'm not...
  23. Austin Chang

    I Prove that V is the internal direct sum of two subspaces

    Let V be a vector space. If U 1 and U2 are subspaces of V s.t. U1+U2 = V and U1 and U1∩U2 = {0V}, then we say that V is the internal direct sum of U1 and U2. In this case we write V = U1⊕U2. Show that V is internal direct sum of U1 and U2if and only if every vector in V may be written uniquely...
  24. binbagsss

    Implicit & explicit dependence derivative sum canonical ense

    Homework Statement Hi, I am trying to follow the working attached which is showing that the average energy is equal to the most probable energy, denoted by ##E*##, where ##E*## is given by the ##E=E*## such that: ##\frac{\partial}{\partial E} (\Omega (E) e^{-\beta E}) = 0 ## MY QUESTION...
  25. X

    (Number theory) Sum of three squares solution proof

    Homework Statement Find all integer solutions to x2 + y2 + z2 = 51. Use "without loss of generality." Homework Equations The Attempt at a Solution My informal proof attempt: Let x, y, z be some integers such that x, y, z = (0 or 1 or 2 or 3) mod 4 Then x2, y2, y2 = (0 or 1) mod 4 So x2 +...
  26. 8

    MHB ACT Problem: Sum Of Even Integers

    What is the sum of all the even integers between 1 and 101? Is there an easier way besides using the formula: (B-A+1)(B+A)/2? It just takes too much time.
  27. VrhoZna

    Proof regarding direct sum of the dual space of a v-space

    (From Hoffman and Kunze, Linear Algebra: Chapter 6.7, Exercise 11.) Note that ##V_j^0## means the annihilator of the space ##V_j##. V* means the dual space of V. 1. Homework Statement Let V be a vector space, Let ##W_1 , \cdots , W_k## be subspaces of V, and let $$V_j = W_1 + \cdots + W_{j-1}...
  28. S

    I How Does Summing Cubic Expansions Reveal the Formula for Sum of Squares?

    I found a deduction to determinate de sum of the first n squares. However there is a part on it that i didn't understood. We use the next definition: (k+1)^3 - k^3 = 3k^2 + 3k +1, then we define k= 1, ... , n and then we sum... (n+1)^3 -1 = 3\sum_{k=0}^{n}k^{2} +3\sum_{k=0}^{n}k+ n The...
  29. MAGNIBORO

    I Solving Riemann Sum Problem: Integral of x^x

    Hi. I try to solve the integral $$\int_{0}^{1} x^{x} dx$$ Through sums of riemann But I came to the conclusion that the result is 0 that is wrong $$\int_{0}^{1} x^{x} dx = \lim_{n\rightarrow \infty }\frac{1}{n}\sum_{k=1}^{n} \left ( \frac{k}{n} \right )^{\frac{k}{n}}$$ $$= \lim_{n\rightarrow...
  30. A

    MHB Can the Sum of Two Unknown Variables be Determined with Limited Information?

    Hi there, I need help with the following situation. Apologies if I'm not using the correct arithmetic terms! Variables: d,e,f e + f = g d / g = j j x e = K j x f = L K + L = M d = M the above situation is a simplified problem, which is easily solvable. Here's where I run into trouble...
  31. Rectifier

    Understanding the Riemann Sum - Integral Connection

    The problem I want to calculate $$\sum^n_{k=1} \frac{4}{1+ \left(\frac{k}{n} \right)^2} \cdot \frac{1}{n}$$ when ##n \rightarrow \infty## The attempt ## \sum^n_{k=1} \underbrace{f(\epsilon)}_{height} \underbrace{(x_k-x_{k-1})}_{width} \rightarrow \int^b_a f(x) \ dx ##, when ##n \rightarrow...
  32. anemone

    MHB What is the Solution to the Complex Sum \sum_{n>1} \frac{3n^2+1}{(n^3-n)^3}?

    Evaluate \sum_{n>1} \frac{3n^2+1}{(n^3-n)^3}.
  33. Albert1

    MHB Sum of 100 terms (continued)

    $a_1,a_2,...,a_{100}\in \begin{Bmatrix} 1,2,3,-----,100 \end{Bmatrix}$ $S=\dfrac{1}{\sqrt{a_1}}+\dfrac{1}{\sqrt{a_2}}+\cdots+\dfrac{1}{\sqrt{a_{100}}}=12.5$. Prove that at least two of the numbers are equal
  34. anemone

    MHB Find the sum of three trigonometric terms

    Evaluate \tan^4 10^\circ+\tan^4 50^\circ+\tan^4 70^\circ without the help of a calculator.
  35. Ryaners

    Finding sum of infinite series: sums of two series together

    Homework Statement Find the sum of the following series: $$ \left( \frac 1 2 + \frac 1 4 \right) + \left( \frac 1 {2^2} + \frac 1 {4^2} \right) +~...~+ \left( \frac 1 {2^k} + \frac 1 {4^k} \right) +~...$$ Homework Equations $$ \sum_{n = 1}^{\infty} \left( u_k+v_k \right) = \sum_{n =...
  36. anemone

    MHB Sum of 100 Terms: Prove At Least 2 Numbers Equal

    The natural numbers $a_1,\,a_2,\,\cdots,\,a_{100}$ are such that $\dfrac{1}{\sqrt{a_1}}+\dfrac{1}{\sqrt{a_1}}+\cdots+\dfrac{1}{\sqrt{a_1}}=20$. Prove that at least two of the numbers are equal.
  37. M

    MHB Cardinality: Is the last sum correct?

    Hey! :o We have the set $A=\{a_1, a_2, \ldots \}$, the $a_i$'s might be finitely or infinitely many. We have that $\mathbb{Q}(A)=\left \{\frac{f(a_1, \ldots , a_n)}{g(a_1, \ldots , a_n)} : f,g\in \mathbb{Q}[x_1, \ldots , x_n], g\neq 0, a_1, \ldots , a_n\in A, n\in \mathbb{N}\right \}$. We...
  38. Ryaners

    Finding sum of infinite series

    [Please excuse the screengrabs of the fomulae - I'll get around to learning TeX someday!] 1. Homework Statement Find the sum of this series (answer included - not the one I'm getting) The Attempt at a Solution So I'm trying to sum this series as a telescoping sum. I decomposed the fraction...
  39. M

    MHB Sum of First 20 Terms of Arithmetic Progression with Even Terms Removed

    First term of the progression is 3 & the common difference is 4 Find the sum of the first 20 terms of the progression that is obtained by removing the terms in the even positions of the given progressions, such as the second term,fourh term, sixth term. Formula preferences For the sum of an...
  40. Albert1

    MHB Finding Minimum Value of $n$ for Given Sum and Product

    $n\in N,\,\,and \,\, a_1,a_2,a_3,-------,a_n\in Z$ $if \,\, a_+a_2+a_3+-----+a_n=a_1\times a_2\times a_3\times------\times a_n=2006$ $find \,\, min(n)$
  41. H

    Total angular momentum is the sum of angular momentum of CM and that about CM

    Consider a flat 2D rigid body rotating about an axis perpendicular to the body passing through a point P that is (1) in the same plane as the body and (2) different from the body's center of mass (CM). In this case does Theorem 7.1 (eqn 7.9) still apply? In the last step of the derivation of...
  42. karush

    MHB 206.8.5.49 Express the integrand as sum of partial fractions

    $\tiny{206.8.5.49}$ $\textsf{Express the integrand as sum of partial fractions}$ \begin{align} && I_{49}&=\int\frac{30s+30}{(s^2+1)(s-1)^3}\, ds& &(1)& \\ &\textsf{expand}& \\ && &=\displaystyle 15\int\frac{1}{(s^2+1)}\, ds -15\int\frac{1}{(s-1)^2}\, ds +30\int\frac{1}{(s-1)^3}\, ds&...
  43. I

    MHB Can we simplify calculating large sums of numbers?

    uhh, how would we get a better way?
  44. D

    Lower Bound on Weighted Sum of Auto Correlation

    Homework Statement Given ##v = {\left\{ {v}_{i} \right\}}_{i = 1}^{\infty}## and defining ## {v}_{n}^{\left( k \right)} = {v}_{n - k} ## (Shifting Operator). Prove that there exist ## \alpha > 0 ## such that $$ \sum_{k = - \infty}^{\infty} {2}^{- \left| k \right|} \left \langle {v}^{\left (...
  45. J

    I Infinite Sum of x/(z+y): Solving the Puzzle

    Can x/(z+y) be written as an infinite sum?
  46. lfdahl

    MHB What is the minimum value of this summation with given constraints?

    Find the minimum of the sum: \[\sum_{i=1}^{5}x_i\], where $x_i \ge 0$ and $\sum_{i<j}|x_i-x_j| = 1.$
  47. B

    How to Sum an Infinite Series?

    Homework Statement Find the sum of the given infinite series. $$S = {1\over 1\times 3} + {2\over 1\times 3\times 5}+{3\over 1\times 3\times 5\times 7} \cdots $$ 2. Homework Equations The Attempt at a Solution I try to reduce the denominator to closed form by converting it to a factorial...
  48. karush

    MHB Ratio Test for Sum $\tiny{206.10.5.84}$

    $\tiny{206.10.5.84}$ \begin{align*} \displaystyle S_{84}&=\sum_{k=1}^{\infty} \frac{(4x)^k}{5k}\\ \end{align*} $\textsf{ ratio test}$ $$\frac{a_{n+1}}{a_n} =\frac{ \frac{(4x)^{k+1}}{5(k+1)}}{ \frac{(4x)^k}{5k}} =\frac{4xk}{k+1} $$ $\textsf{W|A says this converges at $4|x|<1 $ so how??}$
  49. karush

    MHB Sum of Infinite Series: TI and Book Solutions

    $\tiny{206.b.46}$ \begin{align*} \displaystyle S_{book}&=\sum_{k=1}^{\infty} \frac{8^k}{k! }=0\\ S_{TI}&=\sum_{k=1}^{\infty} \frac{8^k}{k! }=e^8-1\\ \end{align*} $\textsf{ 2 different answers?}$
  50. parshyaa

    I Why Is There No Simple Formula for the Sum of a Harmonic Progression?

    What's the reason which implies that we can't have a formula for the sum of HP. https://en.m.wikipedia.org/wiki/Harmonic_progression_(mathematics) Wikipedia gave a reson , can you elaborate it.
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