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. G

    MHB How do you find the sum of an infinite series?

    Hi, I'm trying to solve the sum of following infinite series: \sum_{k=1}^{\infty} \frac{{k}^{2}+4}{{2}^{k}} = \sum_{k=1}^{\infty} \frac{{k}^{2}}{{2}^{k}} + \sum_{k=1}^{\infty} \frac{4}{{2}^{k}} Using partial sum we can rewrite the first series: \sum_{k=1}^{\infty}...
  2. Physics lover

    Sum of a series that tends to infinity

    I tried by ##S=1+(1/1!)(1/4)+(1.3/2!)(1/4)^2+...## ##S/4=1/4+(1/1!)(1/4)^2+(1.3/2!)(1/4)^3..## And then subtracting the two equations but i arrived at nothing What shall i do further?
  3. S

    Understanding Sum to Infinity in Geometric Progression

    My question is Why is the sum to infinity used as opposed to Sum to n? and How can I deduce that the sum to infinity must be used from the question?Total Distance = h + 2*Sum of Geometric progression (to infinity) h + 2*h/3 / 1-1/3 h + 2h/3 *3/2 = h + h = 2h At first I did sum to infinity...
  4. kaliprasad

    MHB Is the Sum of Three Cubes Solved for the Number 33 by a Planetary Supercomputer?

    https://www.quantamagazine.org/sum-of-three-cubes-problem-solved-for-stubborn-number-33-20190326/ has the details
  5. S

    I Showing direct sum of subspaces equals vector space

    If one shows that ##U\cap V=\{\textbf{0}\}##, which is easily shown, would that also imply ##\mathbf{R}^3=U \bigoplus V##? Or does one need to show that ##\mathbf{R}^3=U+V##? If yes, how? By defining say ##x_1'=x_1+t,x_2'=x_2+t,x_3'=x_3+2t## and hence any ##\textbf{x}=(x_1',x_2',x_3') \in...
  6. Joshua G

    Trying to find two vectors with known sum, what am I doing wrong?

    v_1 = <-8/21,-20/21> v_2 = <50/21,-20/21> When I take the dot product of v_2 and <2,5> I get zero, indicating they are perpendicular. Sorry for the hand writing.
  7. lfdahl

    MHB Trigonometric Sum Challenge Σtan^(-1)(1/(n^2+n+1)=π/2

    Show that \[\tan^{-1}(k) = \sum_{n=0}^{k-1}\tan^{-1} \left ( \frac{1}{n^2+n+1} \right ),\;\;\;\;\; k \geq 1,\] - and deduce that \[ \sum_{n=0}^{\infty}\tan^{-1} \left ( \frac{1}{n^2+n+1} \right ) = \frac{\pi}{2}.\]
  8. T

    I Number of Terms for Harmonic Series to Reach a Sum of 100

    I am reading an interesting book by Julian Havil called:" Gamma-Exploring Euler's Constant." Much of the book is devoted to the harmonic series,a slowly diverging series that tends toward infinity.However,one paragraph puzzles me. On p. 23 he says: " In 1968 John W. Wrench Jr calculated the...
  9. DaTario

    Zero Limit of Sum of Squares of Terms with Bounded Range

    I don't know how to show that this limit is zero. It seems that ##\sum_{i=1}^N a_{i,N} /N = 1## and the fact that ## 0 < a_{i,N} < M > 1## implies that some ##a_{i,N}## are less than one. Another conclusion I guess is correct to draw is that ##\lim_{N \to \infty} \sum_{i=1}^N a_{i,N}^2 /N < 1##.
  10. lfdahl

    MHB Maximize the sum of squared distances

    Let $P_i$ denote the $i$thpoint on the surface of an ellipsoid: $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1$, where the principal semiaxes obey: $0 < a < b < c$. Maximize the sum of squared distances: \[\sum_{1\leq i < j \leq 2n}\left | P_i-P_j \right |^2\] - over alle possible...
  11. PainterGuy

    MATLAB Approximating the inverse FT of a unit pulse using a Riemann sum

    Hi, Although I'm using trigonometric form of Fourier transform, first I'd discuss both, exponential and trigonometric forms, for the sake of context. Now proceeding toward the main question and we would only be using trigonometric form. % file name...
  12. J

    Sum of torques, equilibrium problem

    For part a, A force must be applied so that the entire mass can be 'held up'. Therefore the necessary force must be equal to the gravitation force on all the objects: m(rod) * 9.8 + m(LB) * 9.8 + m(RB) * 9.8 = 137 N For part b, (This is where I'm confused) let's set the point at which the...
  13. V

    MHB Computing Kurdyka-Lojasiewicz (KL) exponent of sum of two KL functions.

    Two KL functions $f_1:\mathbb{R}^n\rightarrow \mathbb{R}$ and $f_2:\mathbb{R}^n\rightarrow \mathbb{R}$ are given which have KL exponent $\alpha_1$ and $\alpha_2$. What is the KL exponent of $f_1+f_2$?
  14. R

    I Sum of many random signals

    For instance, here is an example from my own simulations where all underlying signals follow the same analytical law, but they have random phases and amplitudes (such that the sum of the set is 1). The thick line represents the sum: Clearly, the sum tends to progressively get flatter as ##N...
  15. Calculuser

    I Confusion about the Direct Sum of Subspaces

    In "Sheldon Axler's Linear Algebra Done Right, 3rd edition", on page 21 "internal direct sum", or direct sum as the author uses, is defined as such: Following that there is a statement, titled "Condition for a direct sum" on page 23, that specifies the condition for a sum of subspaces to be...
  16. H

    Energy Density of Electric Fields: Is it the Sum of All Electrons?

    I understand that the energy of an electric field arises from the work put into gathering the electrons together to create the field. Bringing electrons close together requires energy because they naturally want to repel. This potential energy is stored in the field itself and the field has an...
  17. Mr Davis 97

    Sum of sides of n polygons in quadrilateral is no more than 4n

    I can construct examples that are less than or equal to ##4n## quite easily, but for the life of me I cannot come with example where it's greater than
  18. Phys pilot

    MATLAB Plot a non homogeneous DPE -- Sum two plots in Matlab?

    Hello, I want to plot this PDE which is non homogeneous: ut=kuxx+cut=kuxx+c u(x,0)=c0(1−cosπx)u(x,0)=c0(1−cosπx) u(0,t)=0u(1,t)=2c0u(0,t)=0u(1,t)=2c0 I have a code that can solve this problem and plot it with those boundary and initial conditions but not with the non homogeneous term...
  19. R

    I'm having difficulty expressing a binomial expansion as a sum

    I found the first 4 terms of the series: ½-(1/16)x^2+(1/64)x^4-(7/1536)x^6. I cannot however simplify this to a sum. the 7 in the numerator of the last term of the above expansion is the sticking point.
  20. bigbob123

    Calc 2 Sum of Alternating Geometric Series

    A0 = 1 A1 = 3 3(An-1) / 4(An-2) = An
  21. S

    I Sum of Binomial Expansion | Spivak Chapter 2, Excercise 3 part d

    Hello, I am working through Spivak for self study and sharpening my math skills. I have become stuck on an exercise. What I need to show is the following: $$ (a + b) \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} = \sum_{j = 0}^{n + 1} \binom{n+1}{j} a^{n-j + 1}b^{j} $$ My attempt, starting from...
  22. S

    Proof by Induction: Arithmetic Sum

    Hi, I am self studying induction and came across the following problem. I am stuck on how to proceed (I need to use induction, I know there is a direct proof). My proof attempt is as follows: Let ## P (m) ## be the proposition that: $$ \sum_{i = m + 1}^{n} i = \frac{(n - m)(n + m + 1)}{2} $$...
  23. lfdahl

    MHB What is the minimum sum of fractions with positive numbers and permutations?

    Let $a_1,a_2, ... , a_n$ be positive numbers. Let $i_1,i_2, ... , i_n$ be a permutation of $1,2,...,n$. Determine the smallest possible value of the sum: $$\sum_{k=1}^{n}\frac{a_k}{a_{i_k}}$$
  24. BWV

    I Proof that the sum of all series 1/n^m, (n>1,m>1) =1?

    Curious about proving that ##\sum_{m=2}^\infty \sum_{n=2}^\infty 1/n^m ## = 1 ran this in Matlab and n,m to 2:1000 =0.9990, and n,m 2:10000 =0.9999, so it does appear to converge to 1
  25. B

    A Vector sum schemes for LS coupling & jj coupling

    The difference between light and very heavy atoms reflects itself in these two schemes. My question is why one scheme for the vector sum is necessarily the right & suitable sum model for one case, and the 2nd scheme suits the 2nd case ? In other words, why & how the relative magnitude of the...
  26. lfdahl

    MHB A positive integer divisible by 2019 the sum of whose decimal digits is 2019.

    Prove the existence of a positive integer divisible by $2019$ the sum of whose decimal digits is $2019$.Source: Nordic Math. Contest
  27. Miles123K

    The sum of this series of the product of 2 sine functions

    Homework Statement I have encountered this problem from the book The Physics of Waves and in the end of chapter six, it asks me to prove the following identity as part of the operation to prove that as the limit of ##W## tends to infinity, the series becomes an integral. The series involved is...
  28. G

    A Binomial as a sum of tetranomials

    Hello there, I'm working on a kinetic theory of mixing between two species - b and w. Now, if I want to calculate the number of different species B bs and W ws can form, I can use a simple combination: (W+B)!/(W!B!) Now, in reality in my system, ws and bs form dimers - ww, bb, wb and bw...
  29. BWV

    I Does the sum of all series 1/n^m, m>1 converge?

    ##\sum_{n=1}^\infty 1/n^2 ## converges to ##π^2/6## and every other series with n to a power greater than 1 for n∈ℕ convergesis it known if the sum of all these series - ##\sum_{m=2}^\infty \sum_{n=1}^\infty 1/n^m ## for n∈ℕ converges? apologies for any notational flaws
  30. lfdahl

    MHB Evaluate the double sum of a product

    Evaluate the following double sum of a product: $$\sum_{j=1}^{\infty}\sum_{n=1}^{\infty}\left(n\prod_{i=0}^{n}\frac{1}{j+i}\right)$$
  31. Demystifier

    Insights The Sum of Geometric Series from Probability Theory - Comments

    Greg Bernhardt submitted a new blog post The Sum of Geometric Series from Probability Theory Continue reading the Original Blog Post.
  32. Spinnor

    I Resultant vector field as sum of many sources

    Let us have some localized density of sources, S, in a plane, each of which produces a localized circular vector field. Let us work in polar coordinates. Let the density of sources, S = Aexp(-r^2/a^2) and let each source have circular vector field whose strength is given by exp(-(r-r_i)^2/b^2)...
  33. CricK0es

    Derivative of a term within a sum

    Homework Statement [/B] From the Rodrigues’ formulae, I want to derive nature of the spherical Bessel and Neumann functions at small values of p. Homework Equations [/B] I'm going to post an image of the Bessel function where we're using a Taylor expansion, which I'm happy with and is as far...
  34. Boltzman Oscillation

    Square of the sum of two orthonormal functions?

    Homework Statement Given: Ψ and Φ are orthonormal find (Ψ + Φ)^2 Homework Equations None The Attempt at a Solution Since they are orthonormal functions then can i do this? (Ψ + Φ) = (Ψ + Φ)(Ψ* + Φ*)?
  35. V

    MHB Square root n limit ( sum question )

    Hi! $$(x_{n})_{n\geq 2}\ \ x_{n}=\sqrt[n]{1+\sum_{k=2}^{n}(k-1)(k-1)!}$$ $$\lim_{n\rightarrow \infty }\frac{x_{n}}{n}=?$$ I know how to solve the limit but I don't know how to solve the sum $\sum_{k=2}^{n}(k-1)(k-1)!$ which should be $(n! - 1)$ The limit would become $\lim_{n\rightarrow \infty...
  36. Y

    MHB Sum of the measures of the interior angles of a heptagon

    PLEASE HELP1.What is the sum of the measures of the interior angles of a heptagon? A. 1260∘ B. 2520∘ C. 900∘ D. 1800∘ my answer is C 5.If the sum of the interior angle measures of a polygon is 3600∘, how many sides does the polygon have? A. 22 sides B. 20 sides C. 18 sides D. 10 sides MY...
  37. R

    I Ways of Getting Sum for n-ary Digits

    For d binary digits, the number of ways W to get sum s is: W(d,s) = d! / (s! * (d-s)!) Are there similar formula(s) for n-ary digits?
  38. opus

    Express this sum as a fraction of whole numbers

    Homework Statement Express the sum as a fraction of whole numbers in lowest terms: ##\frac{1}{1⋅2}+\frac{1}{2⋅3}+\frac{1}{3⋅4}+...+\frac{1}{n(n+1)}## Homework EquationsThe Attempt at a Solution Please see attached image for my work. The reason I am posting the image rather than typing this...
  39. P

    B Quickest way to calculate a given summation

    How would you, personally, do this summation the quickest way?
  40. H

    Equation for sum of torques on a ladder and minimum angle

    Homework Statement [/B] Homework Equations Drawing a diagram for the forces is the easy part. I am not sure I am doing the equation of the sum of the torques well. The Attempt at a Solution This is my attempt for the forces[/B] And this for the torques:
  41. J

    MHB Binomial Sum \displaystyle \sum^{n}_{k=0}\binom{n+k}{k}\cdot \frac{1}{2^k}

    Evaluation of $\displaystyle \sum^{n}_{k=0}\binom{n+k}{k}\cdot \frac{1}{2^k}$
  42. F

    Maximizing Vector Sum with Constant Magnitudes

    <Moderator's note: Moved from a technical forum and thus no template.> Let there be two vectors, u and v. Whose magnitudes are constant u = [a, b] v = [x, y] Define c = ||u|| and k = ||v|| Now sum the vectors: w = u + v = [a, b] +[x, y] = [a+x, b+y] Now find ||w|| ||w|| =√(a+x)2+(b+y)2...
  43. P

    MHB Calculation of probability with arithmetic mean of the sum of random variables

    Calculation of probability with arithmetic mean of random variables There are 4 people, each of whom has one deck of cards with 500 cards that are numbered from 1 to 500 with no duplicates. Each person draws a card from his deck and I would like to calculate the probability of the event that...
  44. M

    MHB Sum of basis elements form a basis

    Hey! :o Let $V$ be a vector space. Let $b_1, \ldots , b_n\in V$ and let $\displaystyle{b_k':=\sum_{i=1}^kb_i}$ for $k=1, \ldots , n$. I want to show that $\{b_1, \ldots , b_n\}$ is a basis of $V$ iff $\{b_1', \ldots , b_n'\}$ is a basis of $V$. I have done the following: Let $B:=\{b_1...
  45. Eclair_de_XII

    Find the density of the sum of two jointly distributed rv's

    Homework Statement "Given that the joint distribution of ##X## and ##Y## is ##f(x,y)=\frac{1}{2}(x+y)e^{-(x+y)},\text { for } x,y>0## and ##0## otherwise, find the distribution of ##Z=X+Y##." Homework Equations ##f_Z(z)=\int_{\mathbb{R}}f(x,z-x)dx##...
  46. K

    Sum of Infinite Series | Calculate the Sum of a Geometric Series

    Homework Statement Find the sum of the series Homework EquationsThe Attempt at a Solution Not sure exactly where to start. If I move 3 outside the sum I'm left with 3*sigma(1/n*4^n), which I can rewrite to 3*sigma((1/n)*(1/4)^n), which party looks like a geometric series..Any tips?
  47. K

    Find the expression for the sum of this power series

    Homework Statement Hello, I need to find an expression for the sum of the given power series The Attempt at a Solution I think that one has to use a known Maclaurin series, for example the series of e^x. I know that I can rewrite , which makes the expression even more similar to the...
  48. S

    MHB Sum of Random Variables...

    Hello I have this following question and I am wondering if i am on the right path : here is the question A picture in which pixel either takes 1 with a prob of q and 0 with a prob of 1-q, where q is the realized value of a r.v Q which is uniformly distributed in interval [0,1] Let Xi be the...
  49. 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...
  50. 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...
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