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. 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...
  2. 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...
  3. AdityaDev

    Summation with binomial coefficients question

    Homework Statement ##\sum\limits_{r=0}^n\frac{1}{^nC_r}=a##. Then find the value of $$\sum\sum\limits_{0\le i<j\le n}(\frac{i}{^nC_i}+\frac{j}{^nC_j})$$ Homework Equations I have used two equations which I derived myself. This is the first one. The second one is: 3. The Attempt at a...
  4. AdityaDev

    Prove that f is a constant function

    Homework Statement Suppose that f:R->R satisfies the inequality ##|\sum\limits_{k=1}^n3^k[f(x+ky)-f(x-ky)]|<=1## for every positive integer k, for all real x, y. Prove that f is a constant function. Homework Equations None The Attempt at a Solution I tried taking f(x)=sinx and then using...
  5. 6c 6f 76 65

    Evaluating the Svein-Graham Sum

    Good evening dearest physicians and mathematicians, I recently came across the so-called "Svein-Graham sum", and i wondered: is it possible to find a simple formula for evaluating it? \sum_{i=0}^k x\uparrow\uparrow i = \left .1+x+x^x+x^{x^x}+ ... +x^{x^{x^{x^{.^{.^{.^x}}}}}}\right \}k
  6. S

    MHB Is the Triangle Inequality Applicable to Norms of Integral Operators?

    Can I always say without reservation that for any two integral operators $K$ and $L$ defined as follows say $(Ky)(x)=\int_{a}^{b} \,k(x,s)y(s)ds$ that $||L||+||K-L||\ge||K||$ thanks Sarrah
  7. AdityaDev

    Proving the Summation Problem: P(x) and the Limit of |e^(x-1)-1| for x>0

    Homework Statement If ##|P(x)|<=|e^{x-1}-1|## for all x> 0 where ##P(x)=\sum\limits_{r=0}^na_rx^r## then prove that ##|\sum\limits_{r=0}^nra_r|<=1## Homework Equations None The Attempt at a Solution ##P(1)=a_0+a_1+...## If the constants are positive, then ##P(1)<=|e^0-1|## So P(1)<=0 so...
  8. S

    M: Solve Riemann Sum Problem Homework

    Homework Statement [/B] Hello, thank you in advance for your help. I am calculating a Riemann sum with right hand endpoints. I hit a small snag, and I appreciate your help in getting me straight.Homework Equations f(x) = x2+ 1, over the interval [0,1]. This is problem number such-and-such from...
  9. Pull and Twist

    MHB Finding the Sum of a Tricky Series

    Find the sum of \sum_{n=1}^{\infty}\frac{1}{n2^{n}} I tried manipulating it to match one of the Important Maclaurin Series and estimate the sum in that fashion but I cannot see to get it to match any. I was thinking of using \sum_{n=1}^{\infty}\frac{\left (\frac{1}{2} \right )^{n}}{n} with the...
  10. Destroxia

    How does a geometric series converge, or have a sum?

    Homework Statement How does a geometric series have a sum, or converge? Homework Equations Sum of Geometric Series = ##\frac {a} {1-r}## If r ≥ ±1, the series diverges. If -1 < r < 1, the series converges. The Attempt at a Solution How exactly does a infinite geometric series have a sum...
  11. R

    Finding the sum of inverse trigonometric expression

    Homework Statement Find ## \sum_1^{23} tan^{-1}(\frac{1}{n^2+n+1}) ## Homework Equations ## tan^{-1}x + tan^{-1}y = tan^{-1}(\frac{x+y}{1-xy} )##The Attempt at a Solution I think we have to split the question in a form of relevant equation given above. First thing what should I do?
  12. anemone

    MHB Find the sum of all possible values

    $a,\,b$ and $c$ are positive real numbers such that $25bc+9ac+ab=9abc$ and $a+b+c=9$. Find the sum of all possible values of $abc$.
  13. Q

    Calculating Infinite Sine Sum with Ratio Test | x and t Real Numbers

    Homework Statement I have this exercise: Calculate ##\sum\limits_{k=0}^\infty t^{k}sin{(kx)}## Where x and t are real and t is between 0 and 1. Homework Equations ? The Attempt at a Solution The ratio test says that this sum does have a limit, and tk obviously converges, as t is between 0 and...
  14. S

    MHB When Does the Norm of the Sum Equal the Sum of Norms for Integral Operators?

    Hi I have 2 linear integral operators $(Ku)(x)=\int_{a}^{b} \,k(x,t)u(t)dt$ $(Mu)(x)=\int_{a}^{b} \,m(x,t)u(t)dt$ I am defining $||K||=max([x\in[a,b]\int_{a}^{b} \,|k(x,t| dt$ same for $L$ when does $||K+M||=||K||+||M||$ thanks sarrah
  15. anemone

    MHB Minimum of the Sum of Logarithms

    Find the minimum of $\large \log_{a_1}\left(a_2-\dfrac{1}{4}\right)+\log_{a_2}\left(a_3-\dfrac{1}{4}\right)+\cdots+\log_{a_n}\left(a_1-\dfrac{1}{4}\right)$ where $a_1,\,a_2,\cdots,a_n$ are real numbers in the interval $\left(\dfrac{1}{4},\,1\right)$.
  16. K

    MHB Direct sum of p-primary components of an R-module counterexample?

    Let $x \in R - \{0\},$ where $R$ is a domain. Define $T_x(M) = \{m \in M \ | \ x^n m=0 \ \ \mathrm{for \ some} \ n \in \mathbb{N}\}$ as the $x$-torsion of $M.$ I know that $T_x(M \oplus N) = T_x(M) \oplus T_x(N)$ for $R$-modules $M,N$ only if $R$ is a PID. But I can't think of a...
  17. K

    MHB Proving that a module can be decomposed as a direct sum of submodules

    Letting $X$ be a ring and $K$ be an $X$-module, I need to show that **if** $K \cong A \times B$ for some $X$-modules $A,B$, **then** $\exists$ submodules $M'$ and $N'$ of $K$ such that: $K=M' \oplus N'$ $M' \cong A$ $N' \cong B.$----------I understand the concepts of internal and external...
  18. S

    Solving Riemann Sum for "Deformation of Water by Magnetic Field

    I'm not sure if this is the right place to post this in, but I'm trying to recreate the "Deformation of water by a magnetic field" experiment by Chen et al. The PDF version of the paper can be accessed via Google (for some reason it won't let me provide a direct link). On the 2nd page of the...
  19. T

    Find the actual sum of a fourier series at a given point

    Homework Statement You have series expansions of the function f(x) = 0 from 0 to .5, and 1 from .5 to 1 : the halfrange cosine series, the half-range sine series, and the Fourier series. For each of these series, find the actual sum of the series at x = 0, and x =1/2, and x =1 Homework...
  20. L

    Sum of exponentials: non-iterative approximate solution?

    Hi everyone, I am trying to solve a problem (related to pharmacokinetics) that requires finding t for an equation like the following: A e^{-a t} - B e ^{-b t} + C e^{-c t} = 0 where A, a, B, b, C and c are all real positive numbers (they are constants related to the absorption, uptake, efflux...
  21. MarkFL

    MHB Solve Trigonometric Sum: Find S Value

    It can be shown that the following sum: S=\sum_{k=1}^{89}\left(\sin^6\left(k^{\circ}\right)\right) is rational. Find the value of $S$. (Callme)
  22. anemone

    MHB Find closed form expression for a given sum

    Find a closed form expression for \sum_{k=1}^{n^2}\dfrac{n-\lfloor\sqrt{k-1}\rfloor}{\sqrt{k}+\sqrt{k+1}}.
  23. carllacan

    How Do Magnetic Planes Influence Current Density in Conductive Materials?

    Homework Statement We have an infinite plane of width 2b made of a magnetic, conducting material (μr >> 1, σ >> 1). Two monochromatic electromagnetic plane waves, with magnetic excitation vector amplitude Hs approach it, each one traveling towards one of its two faces. Find the current density...
  24. alexmahone

    MHB P is the sum of 2 consecutive squares

    Let $p$ be an odd prime. Prove that $p$ is the sum of 2 consecutive squares i.e. $p=a^2+(a+1)^2$ if and only if $p$ has the form $p=\dfrac{u^2+1}{2}$.
  25. F

    How can I evaluate a tricky Euler-Maclaurin sum involving hyperbolic functions?

    Hi everybody, I'm facing a tricky summation problem. The problem is (both mathematically and physically) related to this thread I started a while ago. I'm starting a new thread because the functions are not exactly the same, and I have (perhaps) made some progress, using Euler-Maclaurin sum...
  26. anemone

    MHB Sum of Two Squares: Can $5^{64}-3^{64}$ Be Written?

    Is it possible to write $5^{64}-3^{64}$ as the sum of two squares?
  27. perplexabot

    Double sum of same variable simplification help

    Hello all, I have been meditating on this for a while, but can't seem to understand how this simplification came to be. Any help will be greatly appreciated. So, here is what we start with: ##\mathop{\sum_{k=0}^m\sum_{l=0}^n}_{m{\geq}n} x(k,l)## We also know that: l (lower case L) = n-m+k and...
  28. anemone

    MHB What is the sum of this infinite series?

    Evaluate the infinite series $1-\dfrac{2^3}{1!}+\dfrac{3^3}{2!}-\dfrac{4^3}{3!}+\cdots$.
  29. F

    General Formula for the Reciprocal of a Sum of Reciprocals

    I know that the reduced mass, μ, of an object is: \mu = \frac{1}{\frac{1}{m_1} + \frac{1}{m_2}} \mu = \frac{m_1 m_2}{ m_1 + m_2 } But is there a general formula (or a simplified expression) for finding the value of: \frac{1}{ \frac{1}{x_1} + \frac{1}{x_2} + ... + \frac{1}{x_n} } ? Thank you.
  30. A

    MHB Approximating Position with Riemann Sums

    The question provides a table of values for time and velocity. Part c of the question asks to use a Riemann sum to approximate (not specifying which one). Part d asks what the answer to part c represents and to explain my reasoning. The answer that I got for the sum is 58.5 feet, but I do not...
  31. T

    Sum and difference with radians.

    I think I understand the basic ideas of the sum and difference formulas, I just don't get how to break down say, pi/7 into a form that could be worked with. I could convert it to a degree then back again once I have my answer, but that seems like a lot more work than is necessary. If it were 100...
  32. S

    MHB What does this sum symbol mean?

    $\sum_{i=1}^3 2 i = 12$
  33. E

    Kronecker sum of more than two matrices?

    Homework Statement The question arises from this quote from wikipedia's article about kronecker product: Kronecker sums appear naturally in physics when considering ensembles of non-interacting systems. Let Hi be the Hamiltonian of the i-th such system. Then the total Hamiltonian of the...
  34. M

    MHB Solving for the components of a force at a given angle on a plane.

    Hey! :o We suppose that a force $\overrightarrow{F}$ (for example, the gravity) is applied vertically downwards to an object that is placed at a plane which has an angle of $45^{\circ}$ with the horizantal direction. Express this force as a sum of a force that acts parallel to the plan and of...
  35. W

    Sum of a geometric series of complex numbers

    Homework Statement Given an integer n and an angle θ let Sn(θ) = ∑(eikθ) from k=-n to k=n And show that this sum = sinα / sinβ Homework Equations Sum from 0 to n of xk is (xk+1-1)/(x-1) The Attempt at a Solution The series can be rewritten by taking out a factor of e-iθ as e-iθ∑(eiθ)k from...
  36. S

    MHB Need Sum of Formula [shortcut]

    hi guys. i have 2 questions, how do solve this problem with formula [shortcut] : please, see attachment file.. thanks for your helping.. susanto3311
  37. S

    MHB Fast Calculation [sum] problem

    hello all... would you help me out, how to sum 3 questions problem.. please, see my picture attachment.. thanks for your helping... susanto3311
  38. Sinisa

    Where is sum of divergent series used in physics?

    Hello everyone! My question is: Where is sum of divergent series and divergent integrals used in physics? What it all means? Where can I find examples of divergent integrals? Is there a book of problems for physicists? I am mathematician. I developed a method for summing divergent series...
  39. Suraj M

    Arithmetic progression sum and nth term

    Homework Statement The ratio of sums of 2 AP for n terms each is ## \frac{3n + 8}{7n + 15}## that is $$ {\frac{s_a}{s_b}} = \frac{3n + 8}{7n + 15} $$ find the ratio of their 12th terms. $$ Required= \frac{a₁_a+(n-1)d_a}{a_b + (n-1)d_b}$$Homework Equations Tn = a + (n-1)dThe Attempt at a...
  40. F

    Error propagation for a sum of means

    I have a = {a1, a2, .., a1000}, where this set forms a distribution of photoelectrons (pe) seen by a particular photomultiplier tube (pmt) over 1000 repeated events. I then have N sets of these (N pmts), each containing 1000 pe values which I believe are indeed random and independent. So a, b...
  41. C

    When is the norm of the sum of 2 vectors=sum of norms

    Homework Statement When is |x+y|=|x|+|y| for arbitrary non-zero vectors x,y∈Rn ie, when does equality hold for the well known inequality |x+y|≤|x|+|y| Homework Equations |x|2=<x,x>=Σixi2 |x+y|≤|x|+|y| The Attempt at a Solution squaring both sides of the inequality we have...
  42. K

    MHB Show non-degenerate form of subspaces sum

    Let (E, d) be nonzero bilinear space over K and place conditions: d(x,y) = d(y,x) \\ d(x,y) = - d(y,x) for every x,y \in E. Show that: if E_1 and E_2 are singular (degenerate?) bilinear subspaces relative with d ( (E_1,d|(E_1 \times E_1) and (E_2,d|(E_2 \times E_2) are singular (degenerate?)...
  43. V

    Sum of currents in protection relay

    In a differential protection relay there is a setting of a threshold for the sum of amps (or the squares). In what way would that be of interest? The relay is an Areva P521. Also, the relay can display the ratio between the positive sequence current and the negative. What information does that...
  44. K

    Convert Prod of Sums to Sum of Prod

    Question: Convert to Sum of Products (p+q’+r+s’). (p’+q+r+s’). (p’+q’+r+s’) = ((p+q’+r+s’)’ + (p’+q+r+s’)’ + (p’+q’+r+s’)’)’ (Demorgans each Sum-Term) = ((p’.q.r’.s) + (p.q’.r’.s) + (p.q.r’.s))’ (Factor p.r’.s) = ((p’.q.r’.s) + (p.r’.s).(q’ + q))’ (Inverse Law) = ((p’.q.r’.s) + (p.r’.s))’...
  45. M

    PDE and differentiating through the sum

    Hi PF! I'm reading my math text and am looking at the heat eq ##u_t = u_{xx}##, where we are are given non-homogenous boundary conditions. We are solving using the method of eigenfunction expansion. Evidently we begin by finding the eigenfunction ##\phi (x)## related to the homogenous...
  46. Superposed_Cat

    Given the sum of a series and a term how would you find Tn?

    Hi all, given the sum of a series and a single term how would one find the nth term? Any help appreciated.
  47. J

    What is the output of two cosine functions?

    Homework Statement inputs x1(t) = cos(ω1t), x2(t) = cos(ω2t). Show that output g(t) (sum of x1 + x2) = 0.5cos[(ω2-ω1)t] + 0.5cos[(ω2+ω1)t] Homework Equations included in upload of attempted solution. Trig identities. The Attempt at a Solution Uploaded in pdf. A lot more has been done on the...
  48. M

    Sum of Geometric Series by Differentiation

    Homework Statement Find the sum of the following series: Σ n*(1/2)^n (from n = 1 to n = inf). Homework Equations I know that Σ r^n (from n = 0 to n = inf) = 1 / (1 - r) if |r| < 1. The Attempt at a Solution [/B] I began by rescaling the sum, i.e. Σ (n+1)*(1/2)^(n+1) (from n = 0 to n =...
  49. M

    Sum of n consecutive numbers is divisible by n

    I'm trying to investigate this statement: The sum of n consecutive numbers is always divisible by n. I've found already that it's only true when the total amount of numbers is an odd number. I've also found that the median and mean are the same with consecutive numbers. I can not prove that the...
  50. G

    Is F in P=F/A an average or a sum of all microscopic forces?

    Hello, I've read that the force F we use in the pressure formula P=F/A is an average of all microscopic forces exerted by the gas on the wall of the container over a period of time. Is it true? I thought it was a SUM rather than an average? I assume that in that definition the microscopic force...
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