What is Sums: Definition and 370 Discussions

In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article.
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need of parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one element results in this element itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0.
Very often, the elements of a sequence are defined, through regular pattern, as a function of their place in the sequence. For simple patterns, summation of long sequences may be represented with most summands replaced by ellipses. For example, summation of the first 100 natural numbers may be written as 1 + 2 + 3 + 4 + ⋯ + 99 + 100. Otherwise, summation is denoted by using Σ notation, where






{\textstyle \sum }
is an enlarged capital Greek letter sigma. For example, the sum of the first n natural integers can be denoted as






i
=
1


n


i
.


{\textstyle \sum _{i=1}^{n}i.}

For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find closed-form expressions for the result. For example,







i
=
1


n


i
=



n
(
n
+
1
)

2


.


{\displaystyle \sum _{i=1}^{n}i={\frac {n(n+1)}{2}}.}
Although such formulas do not always exist, many summation formulas have been discovered—with some of the most common and elementary ones being listed in the remainder of this article.

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  1. Euge

    POTW Does the Alternating Binomial Sum Formula Hold for All Positive Integers?

    Show that for all positive integers ##n##, $$\binom{n}{1} - \frac{1}{2}\binom{n}{2} + \cdots + (-1)^{n-1}\frac{1}{n}\binom{n}{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$$
  2. Euge

    POTW Ramanujan Sums: Showing $c_n(k)$

    Consider the ##n##th Ramanujan sum, $$c_n(k) = \sum_{\substack{m = 1\\(m,n) = 1}}^n \exp\left\{2\pi i \frac{k m}{n}\right\}$$ Show that $$c_n(k) = \sum_{d\mid (k,n)} d\, \mu\left(\frac{n}{d}\right)$$
  3. Euge

    POTW Limit of Complex Sums: Find $$\lim_{n\to \infty}$$

    Let ##c## be a complex number with ##|c| \neq 1##. Find $$\lim_{n\to \infty} \frac{1}{n}\sum_{\ell = 1}^n \frac{\sin(e^{2\pi i \ell/n})}{1-ce^{-2\pi i \ell/n}}$$
  4. H

    I Original definition of Riemann Integral and Darboux Sums

    Given a function ##f##, interval ##[a,b]##, and its tagged partition ##\dot P##. The Riemann Sum is defined over ##\dot P## is as follows: $$ S (f, \dot P) = \sum f(t_i) (x_k - x_{k-1})$$ A function is integrable on ##[a,b]##, if for every ##\varepsilon \gt 0##, there exists a...
  5. yucheng

    A References: continuum approximation of discrete sums?

    Is there more references for how accurate is the continuum approximation to discrete sums? Perhaps more mathematical. What I've found: https://lonitch.github.io/Sum-to-Int/ https://arxiv.org/pdf/2102.10941.pdf Some examples are: Sum to integral $$\sum_{\mathbf{k}} \to 2 \left ( \frac{L}{2...
  6. S

    Integral of x^n using Reimann sums

    We don't need to worry about the n = -1 so we can assume that the function is continuous on any interval [a,b] where a, b are real numbers if I separate my interval into N partitions, then the right side values in my interval are a + \frac{b-a}{N}, a + 2 \frac{b-a}{N}, ... , a + k...
  7. mishima

    B Finding all possible sums given 2 lists, matched one to one

    Hi, its been a while since I have thought about this type of math, and I can't really remember how to do this or what its even called. I have two lists of numbers: A: 8, 8, 9, 10, 7, 8 B: 6, 5, 4, 3, 3, 3 I want to find all the different ways I can add elements from A with elements of B. For...
  8. V9999

    I Discrete mathematics--An easy doubt on the notations of sums

    I have a doubt about the notation and alternative ways to represent the terms involved in sums. Suppose that we have the following multivariable function, $$f(x,y)=\sum^{m}_{j=0}y^{j}\sum^{j-m}_{i=0}x^{i+j}$$. Now, let ##\psi_{j}(x)=\sum^{j-m}_{i=0}x^{i+j}##. In the light of the foregoing, is...
  9. C

    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...
  10. rstor

    Exploring Sums of Finite/Infinitesimal Numbers: Q29

    Hello All. This is my first post on the Physics Forums. I have started to self-study calculus and based on the feedback from this site and others, I have chosen Elementary Calculus: An Infinitesimal Approach by Jerome Keisler. I am working through the problems for section 1.5 (page 34/35)...
  11. J

    A Number of unequal integers with sum S

    Hello, I've been trying to solve this problem for a while, and I found a technical solution which is too computationally intensive for large numbers, I am trying to solve the problem using Combinatorics instead. Given a set of integers 1, 2, 3, ..., 50 for example, where R=50 is the maximum...
  12. S

    I Broadly spiralling shape from partial sums of Zeta (0.5 + i t)

    The first plot shows a large number of terms of Zeta(0.5 + i t) plotted end to end for t = 778948.517. The other plots are two zoomed-in regions, including one ending in a Cornu spiral. Despite all sorts of vicissitudes, the plot generally spirals outwards in a "purposeful" sort of way. It is...
  13. K

    MHB How Do You Solve a Geometric Sum with Alternating Signs?

    Hey! I'm stuck again and not sure how to solve this question been at it for a few hours. Any help is appreciated as always. Q: (1) Let the sum S = 3- 3/2 + 3/4 - 3/8 + 3/16 - 3/32 +...- 3/128. Determine integers a , n and a rational number k so that...(Image) (2 )And then calculate S using...
  14. WMDhamnekar

    MHB Expected value and equality to sums

    How to show that$E[N]=\displaystyle\sum_{k=1}^\infty P{\{N\geq k\}}=\displaystyle\sum_{k=0}^\infty P{\{N>k\}}$ If any member here knows the answer, may reply to this question.:confused:
  15. Adesh

    I Checking the integrability of a function using upper and lowers sums

    Hello and Good Afternoon! Today I need the help of respectable member of this forum on the topic of integrability. According to Mr. Michael Spivak: A function ##f## which is bounded on ##[a,b]## is integrable on ##[a,b]## if and only if $$ sup \{L (f,P) : \text{P belongs to the set of...
  16. V

    Python Calculating Riemann Sums on Python w/ Numpy

    import numpy as np def num_int(f,a,b,n): dx=(b-a)/n x=np.arange(a,b,step=dx) y=f(x) return y.sum()*dx def rational_func(x): return 1/(1+x**2) print(num_int(rational_func,2,5,10)) Here is my code for the left endpoint, I know this code works because I compared it to an...
  17. Svein

    Explore the Fascinating Sums of Odd Powers of 1/n

    Continue reading...
  18. S

    Mathematica Will Mathematica Optimize Looping for Partial Sums?

    In this example, DiscretePlot[ Sum[ f[x], {x,1,n} ],{n,1,20}] will Mathematica automatically optimize the procedure -- i.e., will it run a single loop where it calculates the sum up to 20 only once, transferring the partial sums to the output as it goes along? Assume that there is no...
  19. Moni

    A Ascending Order of the Sums of the Elements of the Sub(multi)set of MultiSet

    Summary: Generating Subsets of a Multiset in Ascending Order of the Sums of the Elements of the Sub(multi)set I am trying to come up with an algorithm where you can generate combination from a set in a order such that their sums are in increasing order. This set has to be a multiset i.e...
  20. S

    Riemann sums for discontinuous functions

    The definition of the Riemann sums: https://en.wikipedia.org/wiki/Riemann_sum I'm stuck with a problem in my textbook involving upper and lower Riemann sums. The first question in the problem asks whether, given a function ##f## defined on ##[a,b]##, the upper and lower Riemann sums for ##f##...
  21. WMDhamnekar

    MHB Number of Possible Sums with 1-15 Cards

    There are 8 cards with number 10 on them, 5 cards with number 100 on them and 2 cards with number 500 on them. How many distinct sums are possible using from 1 to all of the 15 cards?Answer given is 143. But my logic is for any sum, at least 2 numbers are needed. So, there are $\binom{15} {2}...
  22. 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...
  23. KF33

    Solving Calculus Homework: Stuck on #11 Riemann's Sum

    Homework Statement I am stuck on number 11 on my homework. Homework Equations Not Sure The Attempt at a Solution I know this has to have something to do with Riemann's Sum, but I am lost on where to start. I started by putting numbers in for p, but I think that is wrong.
  24. D

    I The product of 2 infinite sums

    Hi. I know that eixe-ix = 1 but if I write the product of the 2 exponentials as infinite series I get ΣnΣm xn/(n!) (-x)m/(m!) without knowing the result is 1 using exponentials how would I get the result of this product of 2 infinite sums ? Thanks
  25. R

    Prove the sums of the angles of any given triangle = 180°

    Homework Statement Prove that if A,B,C, are the angles of an arbitrary triangle, then m(A)+m(B)+m(C) = 180 degrees by the following method: From any vertex draw the perpendicular to the line of the opposite side. Then use the result already known for right triangles Homework EquationsThe...
  26. tomdodd4598

    I Exploring Direct Sums of Lorentz Group Representations

    Hey there, I've suddenly found myself trying to learn about the Lorentz group and its representations, or really the representations of its double-cover. I have now got to the stage where the 'complexified' Lie algebra is being explored, linear combinations of the generators of the rotations...
  27. Glenn Rowe

    A Polarization vector sums in QED

    I'm working through Lahiri & Pal's book A First Book of Quantum Field Theory, Second Edition and I'm stuck on their explanation of the polarization vector in quantum electrodynamics in Chapters 8 and 9. In section 8.8, they derive a formula for the sum over the transverse polarization modes of...
  28. 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...
  29. P

    Calculus 2 for Engineers: Riemann sums

    Homework Statement a. Write down a Riemann sum for the integral ∫x3dx from 0 to 1. b. Given the following identity 13+23+33...+N3=(N(N+1)/2)2, show that the Riemann sums for ∫x3dx from 0 to 1 converge to 1/4. The Attempt at a Solution I believe I have gotten part a. I got ∑i^3/N^4 from i=0 to...
  30. T

    MHB Definite integral with Riemann sums

    Heya, So, I know this is a pretty simple problem, but I seem stuck on it nevertheless. Here's the question Calculate the upper and lower sums , on a regular partition of the intervals, for the following integrals \begin{align*} \int_{1}^{3}(1-7x)dx \end{align*} Please correct me if I'm doing...
  31. Mr Davis 97

    Showing that partial sums diverge to infinity

    Homework Statement Let ##\sum_{n=1}^{\infty}a_n## be a series with nonnegative terms which diverges, and let ##(s_n)## be the sequence of partial sums. Prove that ##\lim_{n\to\infty} s_n = \infty##. Homework EquationsThe Attempt at a Solution This isn't a difficult problem, but I want to make...
  32. J

    MHB Real Analysis, Sequences in relation to Geometric Series and their sums

    I will state the problem below. I don't quite understand what I am needing to show. Could someone point me in the right direction? I would greatly appreciate it. Problem: Let p be a natural number greater than 1, and x a real number, 0<x<1. Show that there is a sequence $(a_n)$ of integers...
  33. Math Amateur

    MHB Understanding Bland's Proposition 4.2.10 in Rings and Modules

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  34. Math Amateur

    MHB Noetherian Modules: Direct Sums & Bland Proposition 4.2.7

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some further help to fully understand the proof of part of Proposition 4.2.7 ... ... Proposition 4.2.7 reads as...
  35. Math Amateur

    MHB Direct Sums of Noetherian Modules .... Bland Proposition 4.2.7 .... ....

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.7 ... ... Proposition 4.2.7 reads as follows:https://www.physicsforums.com/attachments/8208In...
  36. mnb96

    A Does Closure Under Multiplication in One Subspace Imply the Same for Another?

    Hi, consider a (finite dimensional) vector space ##V=U\oplus W##, where the subspaces ##U## and ##V## are not necessarily orthogonal, equipped with a bilinear product ##*:V\times V \rightarrow V##. The subspace ##U## is closed under multiplication ##*##, thus ##U## is a subalgebra of ##V##...
  37. Math Amateur

    MHB Direct Sums and Factor Modules .... Bland Problem 14, Problem Set 2.1

    I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ... I need help to make a meaningful start on Problem 14 of Problem Set 2.1 ... Problem 14 of Problem Set 2.1 reads as follows:I am somewhat overwhelmed by...
  38. Math Amateur

    I External Direct Sums and the Sum of a Family of Mappings ....

    I have an issue/problem that relates to Bland initial treatment of external direct sums including Proposition 2.1.5 ... especially Bland's definition of the sum of a family of mappings ... Bland's text on this is as follows: In the above text by Bland we read the following: " ... ... We now...
  39. Math Amateur

    MHB Short Exact Sequences and Direct Sums .... Bland, Proposition 3.2.7 .... ....

    I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 3.2 Exact Sequences in ModR ... ... I need some help in order to fully understand the proof of Proposition 3.2.7 ... Proposition 3.2.7 and its proof read as follows...
  40. Math Amateur

    I Short Exact Sequences & Direct Sums .... Bland, Proposition 3.2.7

    I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 3.2 Exact Sequences in ##\text{Mod}_R## ... ... I need some help in order to fully understand the proof of Proposition 3.2.7 ... Proposition 3.2.7 and its proof read as follows: In the above...
  41. Math Amateur

    MHB External Direct Sums and Direct Products .... Bland Problem 1, Section 2.1 ....

    I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ... I need help with Problem 1(b) of Problem Set 2.1 ... Problem 1(b) of Problem Set 2.1 reads as follows: I have had difficulty in formulating a rigorous...
  42. Math Amateur

    External Direct Sums & Direct Products .... Bland Ex. 1b, 2.1

    Homework Statement I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ... I need help with Problem 1(b) of Problem Set 2.1 ... Problem 1(b) of Problem Set 2.1 reads as follows:Bland Problem 1, Section 2.1...
  43. MountEvariste

    MHB Sum of powers limit via Riemann sums?

    One of the many excellent problems by lfdahl in the challenge questions and puzzles subforum was recently: https://mathhelpboards.com/challenge-questions-puzzles-28/prove-limit-23480.html My first idea was Riemann sums! I didn't succeed. So I ask, can this limit be calculated via Riemann...
  44. Euler2718

    Limit of Partial Sums involving Summation of a Product

    Homework Statement Show that the sequence of partial sums s_{n} = 1+\sum_{i=1}^{n} \left(\prod_{k=1}^{i}\left( \frac{1}{2} + \frac{1}{k}\right)\right) converges, with n\in \mathbb{N}\cup \{0\} Homework EquationsThe Attempt at a Solution [/B] So we want to find \lim_{n\to\infty} s_{n} =...
  45. Svein

    Insights Further Sums Found Through Fourier Series - Comments

    Svein submitted a new PF Insights post Further Sums Found Through Fourier Series Continue reading the Original PF Insights Post.
  46. lfdahl

    MHB Evaluating $(-1)^k {3n \choose k}$ Sums

    Evaluate the sum: $$S_n =\sum_{k=0}^{n}(-1)^k{3n \choose k}, \;\;\;n=1,2,...$$
  47. P

    MHB Dale's questions via Facebook about Riemann Sums

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  48. G

    I Divergent Sums of Linearly Independent Elements

    Suppose we had an infinite series - z = ∑i = 1 to ∞ ( α1(i)x1 + α2(i)x2 + . . . + αm(i)xm ) - rewritten as the cumulative sequence - z(n) = α1(n)x1 + α2(n)x2 + . . . + αm(n)xm - where the xj are linearly independent and normalized (and serve as a finite basis across the sequence). If all...
  49. Yiming Xu

    I Express power sums in terms of elementary symmetric function

    The sum of the $k$ th power of n variables $\sum_{i=1}^{i=n} x_i^k$ is a symmetric polynomial, so it can be written as a sum of the elementary symmetric polynomials. I do know about the Newton's identities, but just with the algorithm of proving the symmetric function theorem, what should we do...
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