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. H

    Finding the sum of an indefinite series

    Homework Statement Ʃ(x-->infinity, x>0) 11/(n(n+2)) Homework Equations The Attempt at a Solution I am not quite sure what to do, i think that i am supposed to put it into partial fractions. i changed it to the form 11/(n^2+2n) --> 11/((n+1)(n+1)-1)) --> 11/((n+1)^2-1)...
  2. M

    What is the solution to this simple sum problem?

    (1/1-1/6) + (1/2-1/7) + (1/3-1/8)...+(1/95-1/100) This was on the cover of a local paper with the caption "You can't solve this but he can!" The 'he' in the caption was a 12 year old boy. On the next page they gave as his solution 2.2. Two things are clear, 1.-the boy had the right answer and...
  3. G

    Euler sum of positive integers = -1/12

    My question arises in the context of bosonic string theory … calculating the number of dimensions, consistent with Lorentz invariance, one finds a factor that is an infinite sum of mode numbers, i.e. positive integers … but it really goes back to Euler, and his argument that the sum of all...
  4. A

    All possible ways to sum to a number

    I am curious if there is a universal formula to find all possible sums of a given number. For instance, to add to 10: 1+9 2+8 1+1+8 2+2+2+3+1, etc I came up with a simple algorithm, but I'm sure there is something similar to Gauss's formula which can be utilized. I have heard Partitions...
  5. T

    Subsequence that Sums Up to Half the Total Sum

    Hi all, I was just looking at the U.S. electoral map, and I was wondering if there could possibly be a tie in presidential elections (the answer is probably no). I tried to think of an efficient algorithm to answer this question, but due to my limited intelligence and imagination, all I...
  6. N

    Solving Combinatorics Sum: Feyman's Logic Problem

    Hi all, Can anyone gimme any clues to solve the sum below (or solve it outright :D)? \sum_{i=k}^{n} \frac{i!}{(i-k)!} I'm trying to solve one of Feyman's logic problems (bored geek alert) and I'm stuck at this point. And since my high school days are so far behind... Thanks in...
  7. D

    Partial Sum Approximation for Alternating Harmonic Series

    Homework Statement Find a value for n for which the nth partial sum is ensured to approximate the sum of the alternating harmonic infinite series to three decimal places. Homework Equations Sn = Ʃ(-1)^k+1*1/k = 1 - 1/2 + 1/3 - 1/4 + 1/5 - . . . S1 = 1 S2 = 1 - 1/2 S3 = 1 - 1/2 + 1/3 S4...
  8. R

    Proof the identities of the sine and cosine sum of angles

    Homework Statement I just have to prove the well known identities: \cos(\alpha + \beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta) \sin(\alpha + \beta)=\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin( \beta) But the thing is that I've to use the Taylor power series for the sine and cosine...
  9. R

    Finding Extrema of Sum of Three Sines

    Hello everybody, I'm new to this forum so thanks for having me. I'm trying to find the times when the extrema occur for a periodic wave f(t) equal to the sum of three sine waves. Given f(t) = sin(2∏at) + sin(2∏bt) +sin(2∏ct) where a, b and c are whole numbers in lowest form (i.e...
  10. D

    Finding which direct sum of cyclic groups Z*n is isomorphic to

    I always see problems like "how many structurally distinct abelian groups of order (some large number) are there? I understand how we apply the theorem which tells us that every finite abelian group of order n is isomorphic to the direct sum of cyclic groups. We find this by looking at the...
  11. D

    Sum of alternating series using four-digit chopping arithmetic

    Homework Statement Let a_{n} be an alternating series whose terms are decreasing in magnitude. How to compute the sum as precisely as possible using four-digit chopping arithmetic? In particular, apply the method to compute \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}}}{{(2n)!}}} and...
  12. alyafey22

    MHB Proof the convergence of a gamma sum

    How to prove the convergence or divergence of ? $$\sum^{\infty}_{n=1}\frac{\Gamma(n+\frac{1}{2})}{n\Gamma{(n+\frac{1}{4})}}$$
  13. B

    MHB Internal angle sum of triangle

    Problem: Let A, B, C be three non-collinear points. Let D, E, F be points on the respective interiors of segments BC, AC and AB. Let θ, φ and ψ be the measures of the respective angles ∠BFC, ∠CDA and ∠AEB. Prove IAS(ABC) < θ +φ + ψ < 540 - IAS(ABC).(IAS means internal angle sum). Now I am...
  14. anemone

    MHB Rationalizing a denominator involving the sum of 3 cube roots

    Hi members of the forum, Problem: Rationalize the denominator of $\displaystyle \frac{1}{a^\frac{1}{3}+b^{\frac{1}{3}}+c^{\frac{1}{3}}}.$ I know that if we are asked to rationalize, say, something like $\displaystyle \frac{1}{1+2^{\frac{1}{3}}}$, what we could do is the following...
  15. A

    Sum of Second Order Linear PDEs

    Suppose we have two multivariate functions, u_{1}(x,t) and u_{2}(x,t). These functions are solutions to second-order linear equations, which can be written as follows: Au_{xx}+Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+Fu=G Each of the coefficients are of the form A(x,y). Now, the linearity of these...
  16. P

    Determining convergence of a sum

    I'd really appreciate some help with a sum of: a_n= |sin n| / n All I've thought of, is that I should probably create a subsequence of {a_n}, such that all the elements of this subsequence {a_n_k} are >epsilon >0, and then compare the subsequence to 1/n which diverges. However, I have no...
  17. Biosyn

    Sum of 5^1-5^2+5^3-5^4+...-5^{98}: e. (5/6)(1-5^98)

    Homework Statement Find the sum of 5^1-5^2+5^3-5^4+...-5^{98} a. (5/4)(1-5^99) b. (1/6)(1-5^99) c. (6/5)(1+5^98) d. (1-5^100) e. (5/6)(1-5^98) Homework Equations The Attempt at a Solution I feel as though this is actually a simple problem and that I'm not looking at it the right way. [5^1...
  18. tsuwal

    How to Calculate 3^2048: Step-by-Step Guide

    the answer is 3^2048. How do I get there?
  19. S

    Computing the sum of a particular series.

    Homework Statement The Attempt at a Solution Alright, so, I'm clueless about doing this one. I do know that it's extremely similar to e^x \sum_{n->0}^{\infty}\frac{x^n}{n!} But really, that means nothing! Usually there's another function/series I can compare and then integrate, in...
  20. R

    Getting the components of a sum of square waves

    If I have a periodic function that is say a sum of a number of sine functions I can use a Fourier Transform to get the component functions. Now, if I have a bunch of square waves of differing amplitudes and frequencies that I add up into a resultant waveform. Given that waveform what'd be...
  21. anemone

    MHB What is the sum of α + β in these given equations?

    Hi members of the forum, Problem: The real numbers $\displaystyle \alpha$, $\displaystyle \beta$ satisfy the equations $\displaystyle \alpha^3-3\alpha^2+5\alpha-17=0,$ $ \displaystyle \beta^3-3\beta^2+5\beta+11=0.$ Find $\displaystyle \alpha + \beta$. This problem has me stumped. It's...
  22. 0

    What is the lowest sum for K=2 in this infinite sequence with specific criteria?

    Homework Statement Say we have an infinite sequence of natural numbers A such that no K subsequences can be found adjacent such that the average of the elements in any subsequence is equal for all K subsequences. Sorry about my poor description, an example would be that {2, 3, 4, 1} wouldn't...
  23. B

    Sum of work = Work done by net force

    Suppose you have many forces acting on an object, and the object moves in space in some time interval. Each force has done some work on the object. Suppose you took all these values for work, added them up, (they are all scalars). You'd obtain a scalar equal to the net work done on the object...
  24. L

    Express e^x from 1 to 8 as a Riemann Sum. Please, check my work?

    Express e^x from 1 to 8 as a Riemann Sum. Please, check my work? :) 1. Express ∫1 to 8 of e^xdx as a limit of a Riemann Sum. (Please ignore the __ behind the n's. The format is not kept without it...) _____n 2. lim Ʃ f(xi)(Δx)dx x→∞ i=1 Δx= (b-a)/n = 8-1/n = 7/n xi= 1 + 7i/n ____n lim Ʃ...
  25. M

    Riemann Sum with subintervals/partition

    So I missed a class and am trying to figure out a question in my textbook but am completely lost. It goes a little something like this: Let f(x)=x3 and let P=<-2,0,1,3,4> be a partition of [-2,4]. a) Compute Riemann Sum S(f,P*) if the points <x1*,x2*,x3*,x4*>=<-1,1,2,4> are embedded in P...
  26. Jameson

    MHB Prove sum of (-1)^i times n choose i equals 0

    Problem: Prove that for $n>0$, \sum_{i=0}^{n} (-1)^i \binom{n}{i}=0 Attempt: This seems clearly like a proof based on induction. 1) Base case: for $n=1$, \sum_{i=0}^{1}(-1)^i \binom{1}{i}=(-1)^0 \binom{1}{0}+(-1)^1 \binom{1}{1}=1-1=0 2) Show that $n=k$ being valid implies $n=k+1$ is valid...
  27. M

    Sum problem about closing form

    \sum_{i=1}^{n} |(y_i-\theta)|=n\theta where theta is a fixed constant and y_i is a discrete random variable. does anyone know how to rewrite in close form?? also, everytime i use latex it starts a new line. how can i fix this so i can type directly with my sentances? thanks
  28. U

    Find the sum to infinite series

    Homework Statement cot^-1 3 + cot^-1 7 + cot^-1 13+... Homework Equations The Attempt at a Solution I first tried to write the nth term of the series t_n = cot^{-1}\left( 2^n + (2n-1) \right) Then I tried to calculate the limit as n→∞. But I simply can't do that. I mean I...
  29. J

    MHB Sum of first m terms of a combinatorial number

    Dear Math Help Boards, I have a tricky problem that I hope one of you can help me with. (It's for a personal project, nothing to do with school.) I'm looking for a closed-form expression for the sum of the first through m-th terms of a combinatorial number. For those of you unfamiliar with...
  30. J

    Closed form expression for sum of first m terms of a combinatorial number

    Dear Physics Forums denizens, I have a tricky problem that I hope one of you can help me with. (It's for a personal project, nothing to do with school.) I'm looking for a closed-form expression for the sum of the first through m-th terms of a combinatorial number. For those of you...
  31. P

    Finding the Sum of a Power Series: Tips and Tricks for Success

    Homework Statement I am trying to find the sum of the series in the attachment. Homework Equations The Attempt at a Solution I have tried to use various series and their derivatives, to not much avail. I am not sure how to handle the n^2 factor. Should I break it down to two...
  32. M

    Happy New Year: Infinite Sum Question Explanation

    Happy new year. All the best. I have one question. Is it true? \sum^{\infty}_{k=0}a_kx^k=\sum^n_{k=0}a_{n-k}x^{n-k} I saw in one book relation \sum^{\infty}_{k=0}\frac{(2k)!}{2^{2k}(k!)^2}(2xt-t^2)^k=\sum^{n}_{k=0}\frac{(2(n-k))!}{2^{2(n-k)}((n-k)!)^2}(2xt-t^2)^{n-k} Can you give me some...
  33. R

    Solving Fractional Part Sum S(n): a,b,n Natural Non-Null Numbers

    Hi everyone! How to solve this: S(n) = { (a+b)/n } + { (2a+b)/n } + { (3a+b)/n } + ... + { (na+b)/n } where {x} represents fractional part of x. a,b,n are natural non-null numbers and (a,n)=1. I don`t need only an answer, i need a good solution. Thanks!
  34. U

    What is the sum of complex solutions?

    Homework Statement Let z1, ... zn be the set of n distinct solutions to the equation zn = a where a is a complex number. (a) By considering distinct solutions as the sides of a polygon in an Argand diagram show that these sum to zero.(b)...
  35. A

    When can interchange sum and integral

    Why in the attached picture is it legal to interchange the sum and integral? Is it just because n is not dependent on t? note: (c1)n is just a function of n
  36. S

    Nonlinear Systems & Weighted Sum of Impulses

    Hello, my question is that almost all textbooks say that a linear system will give the output to a weighted sum of impulses which equals the superposition of scaled responses to each of the shifted impulses. But if we apply the same input which is a weighted sum of impulses to a non linear time...
  37. DryRun

    Evaluate Sum of $$2(x_m-x_0)-3(y_n-y_0)$$ Homework

    Homework Statement $$\sum_{i=1}^{m} \sum_{j=1}^{n}[2(x_i-x_{i-1})-3(y_j-y_{j-1})]$$ Homework Equations Multiple-sigma notation.The Attempt at a Solution I agree this seems like basic summation stuff, but i do not agree with the given answer. So, here is what I've done. $$\sum_{i=1}^{m}...
  38. suyver

    Is the Mean of a Sum of Randomly Chosen Numbers Always 1?

    I choose a random number p_1 \in [0,1) and a subsequent series of (increasingly smaller) random numbers p_i \in [0, p_{i-1}). Then I can calculate the sum \sum_{i=1}^\infty p_i. Naturally, this sum is dependent on the random numbers chosen, so its particular result is not very insightful...
  39. B

    Evaluating the sum of a sigma notation problem with a lower limit k=10

    How do I evaluate the sum of this sigma notation problem? 20 ∑ k k=10 Normally, I would think to use the theorem for the sum of the first n integers: n ∑ k = n(n+1)/2 k=1 I don't know how to do this, however, since the lower limit is k=10, not k=1. My professor wrote this note on the board...
  40. T

    Complex number sum that should be easy

    Hey, So I have a sum of complex numbers that really should be easy, but I'm not getting the right solution. It is with respct to using the Gram Schmidt process U1 = (i, -1, i) U2 = (1,1,0) So I perform the Gram Schmidt with U1 being my initial vector selection and I get: V2 =...
  41. M

    Need help proving that an infinite double sum is 1

    Homework Statement I am asked to prove that e^{iB} is unitary if B is a self-adjoint matrix. The Attempt at a Solution In order to prove this I am attempting to show e^{iB} \widetilde{e^{iB}} = 1. Using the assumption that B is self-adjoint I have been able to show that e^{iB}...
  42. P

    Sum of two subspaces - question.

    Homework Statement Is it possible to add the following subspaces: W_1 = Sp{(1,0,0)} and W_3 = Sp{(0,1,-1), (0,0,1)}? Homework Equations The Attempt at a Solution Will their sum be: Sp{(1,1,-1),(1,0,1)}?
  43. R

    Sum of Torque vs. Conservation of Angular Momentum Quick help needed

    Number 11) This is what I did: Ʃτ = F2r2 + F1r1 = 0 (195)(7) + F1(0.7) = 0 F1(0.7) = -(195)(7) F1 = -1365/0.7 F1 = -1950 N F1 = 1950 N Is that answer right? Number 12) This is what I did: Since this is a massless rod and the location of the axis is through the end, I = ML2 Linitial =...
  44. Y

    Sum of N geometric variables with changing probability

    Homework Statement Ʃ(A-i)/(N+1-i) sum of i=1 to N Homework Equations How do I solve this series for all 0<N<A cases. This series is the sum of N geometric variables of changing probability. I'd appreciate any help
  45. M

    Expressing the limit of a sum as a definite integral

    Homework Statement Express the following as a definite integral: Express the attached limit as an integral. The Attempt at a Solution I have gotten as far as every part of the answer except the upper bound. the answer is: 10 ∫(from 1 to 10) [x-4lnx]dx 1 since the definition of...
  46. E

    Roots of linear sum of Fibonacci polynomials

    For what complex numbers, x, is Gn = fn-1(x) - 2fn(x) + fn+1(x) = 0 where the terms are consecutive Fibonacci polynomials? Here's what I know: 1) Each individual polynomial, fm, has roots x=2icos(kπ/m), k=1,...,m-1. 2) The problem can be rewritten recursively as Gn+2 = xGn+1 +...
  47. S

    Basic linear algebra direct sum questions

    Homework Statement I'm reading from the first edition of Axler's Linear algebra done right. In the section on sums of vector subspaces, he states: U = {(x,0,0) ∈ F3 | x ∈ F} W = {(y,y,0) ∈ F3 | y ∈ F} and 1.7 U + W = {(x,y,0) ∈ F3 | x,y ∈ F} However, shouldn't the answer be U...
  48. D

    MHB Exploring the Sum of $\frac{1}{\sqrt{1 + n^2} + n}$

    $\sum\limits_{n = 1}^{\infty}\left(\sqrt{1 + n^2} - n\right)$ $$ \sqrt{1 + n^2} - n = \frac{1}{\sqrt{1 + n^2} + n} $$ Now what?
  49. D

    MHB Sum of an Infinite Series with Real Exponent p

    $\sum\limits_{n = 2}^{\infty}n^p\left(\frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}}\right)$ where p is any fixed real number. If this was just the telescoping series or the p-series, this wouldn't be a problem.
  50. N

    Calculate 5Σ r=0 r(r+1): Find the Sum!

    Calculate 5 \Sigma r=0 r(r+1) (Sorry I don't know how to do the proper notation online) How do you calculate the sum for this since the common difference is changing? I tried to write them out separately so the sum of r x sum of r+1 but I don't know how you put that in the sum formula. Sn =...
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