Sums Definition and 370 Threads

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. K

    Representing numbers as sums of fibonacci numbers

    I have the following homework to do. Apologies if it seems very easy - I just had a knee surgery and I think I can't really think straight due to pain medication, I feel so fuzzy and sleepy and my damn knee still hurts like $#%&. So instead of representing numbers in a binary way I need to...
  2. T

    Small lemma about sums and products

    Hey, I'm trying to prove a larger theorem; in order to complete my proof I need to use the following lemma (or, if it turns out to be false, try a completely different method of proof): Consider any two sets of n nonzero integers, A and B. If their respective sums and products are equal...
  3. H

    Convergent Series and Partial Sums

    Homework Statement Let \sum_{n=1} a_n and \sum_{n=1} b_n be convergent series. For each n \in \mathbb{N}, let c_{2n-1} = a_n and c_{2n} = b_n. Prove that \sum_{n=1} c_n converges. Homework Equations The Attempt at a Solution Not sure whether the following solution is...
  4. V

    (Nevermind) Establish Trig Identity: Sums to Products

    Homework Statement Establish the identity: 1+cos(2θ)+cos(4θ)+cos(6θ)=4cosθcos(2θ)cos(3θ) Homework Equations cos(a)+cos(b)=2cos((a+b)/2)cos((a-b)/2) The Attempt at a Solution I understand how to do a simple cos(+/-)cos problem according to the Sums as Products equations, but I am...
  5. J

    Proving Binomial Sums: Step-by-Step Guide for Solving Homework Equations"

    Homework Statement https://www.physicsforums.com/attachment.php?attachmentid=39642&stc=1&d=1317853920 how do you go about solving this? Homework Equations i have proved the binomial theorem.The Attempt at a Solution i was considering cases, for j(even or odd). would this be the right direction?
  6. A

    Question related to Riemann sums, sups, and infs of bounded functions

    Can someone give me an example of a bounded function f defined on a closed interval [a,b] such that f does not attain its sup (or inf) on this interval? Obviously, f cannot be continuous, but for whatever reason (stupidity? lack of imagination?) I can't think of an example of a discontinuous...
  7. 1

    Riemann Sums and Integrals, feel lost without actual functions

    Homework Statement At my old university, Calculus was taught much differently than it is where I am now. My old school focused on numerical things, which this school focuses much more on pictures, abstract, etc. and it's very difficult for me. At my old school, we were given a shape...
  8. B

    Statistical mechanics: Sums of exponentials with sums.

    Homework Statement I'm working through an example from class and the textbook, but I'm confused about how the steps progress mathematically. The example involves the Gibb's partition for a paramagnet. \sum_{s} exp(\beta \mu B \sum_{i}^{N} s) Where s = -a,-a+1...a for each spin...
  9. C

    Infinite riemann sums discrepancy

    Hello. I have to solve some integrals using both the standard theorem of calculus and infinite Riemann sums. \int_{1}^{7} (x^2-4x+2) dx = \lim_{n \to \infty } \sum f(x_i)\Delta x_i = \lim_{n \to \infty } \sum (x_i^2 - 4x_i + 2)6/n Evaluating the definite integral results in an answer of 30...
  10. T

    At what point do the sums of the reciprocals converge?

    It is well known that the Harmonic Series diverges (1/1+1/2+1/3+1/4+...), but that the series [1/1+1/4+1/9+1/16+...] converges. In the first series, the denominators are the integers, whereas in the second example, the denominators are the integers to the power of 2. My question is, at what...
  11. R

    Can Riemann sums accurately calculate the area under a curve?

    1) if you have 2 line segments making a right angle, and connect the endpoints with a line segment with an outward curve relative to the vertex, will the area inside always be irrational? 2) if you have 2 line segments making a right angle, and connect the endpoints with a line segment with an...
  12. A

    How Can You Derive the Formula for the Sum of the First n Integers?

    Hello, i just came accros: Sum(i) , from i=1 to i=n which apparently equals n(n+1)/2 -Is there a way to derive this from the sum, or you just have to use your intuition and think through what exactly is being summed and the range of summation? -Do you have any resources to offer, that...
  13. T

    Linear Algebra: quick little question about sums of subspaces

    Homework Statement Learning about sums of subspaces and wanted to be sure that I am understanding this correctly. Say that I have two subspaces of R^2: U = {(x,y) in R^2 : y + 2x = 0} W = {(x,y) in R^2 : y - 3x = 0} and I wanted to geometrically (and algebraically) represent their...
  14. C

    Geometric series partial sums question

    I am looking at a geometric series problem that has already been worked out, so not assigned, but I do not see where they get a number: Summation from n=1 to inf: 1/(n^2+4n+3) In doing the partial sums, he has (1/2)* summation... 1/(i+1) - 1/(i+3) I understand the breakup, but where does...
  15. D

    Sums and products of random variables

    Can anyone help me with the below question? for each of the following pairs of random variables X,Y, indicate a. whether X and Y are dependent or independent b. whether X and Y are positively correlated, negatively correlate or uncorrelated i. X and Y are uniformly distributed on the disk...
  16. D

    Probability: Sums and Products of Random Variables

    Homework Statement Suppose that X is uniformly distributed on (0,2), Y is uniformly distributed on (0,3), and X and Y are independent. Determine the distribution functions for the following random variables: a)X-Y b)XY c)X/Y The Attempt at a Solution ok so we know the density fx=1/2...
  17. B

    Show that if all the row sums of a matrix A belong to C (nxm) are

    show that if all the row sums of a matrix A belong to C (nxm) are zeroes, then A is singular. Hint. Observe that Ax=0 for x=[1 1 ...1]T
  18. F

    Use a left and midpoint Rienman sum to approximate

    Homework Statement Use a left and midpoint Rienman sum to approximate \int_{0}^{3} x^3 dx The Attempt at a Solution Left \Delta x = \frac{3}{n} Now here is the problem, for left points, what is my 'a'? I can't carve up my intervals and this isn't a right-end approximation so I...
  19. F

    Mr.Stewart is confusing me. Estimating sums

    Homework Statement http://img842.imageshack.us/img842/1404/unledro.png Now what I don't understand is this part [PLAIN][PLAIN]http://img821.imageshack.us/img821/7479/unledoq.png Is he trying to say that \frac{1}{2n^3} = R_n? Is that how he deduced that \frac{1}{2n^3} < 0.0005?
  20. K

    Riemann Sums: Finding the Limit as n Approaches Infinity

    Homework Statement Identify an=the summation from k=1 to n of (2n)/(4k2+1) as a Riemann sum of an appropriate function on an appropriate interval and find the limit as n approaches infinity of an. Homework Equations There is no interval givien so I assume its from 0 to 1. The...
  21. M

    Show that the partial sums of a power series have no roots in a disk as n->infty

    Homework Statement Let f_n(z)=\sum_{k=0}^n\frac{1}{k!}z^n. Show that for sufficiently large n the polynomial f_n(z) has no roots in D_0(100), i.e. the disk of radius 100 centered at 0. Homework Equations This is a sequence of analytic functions which converges uniformly to e^z on C...
  22. B

    Converting between Sums of Products & Products of Sums

    Homework Statement Translate the following equation into canonical S of P form: F = (xyz' + x'w)(yz + x'z') Homework Equations DeMorgan's theorem? (x + y)' = x'y' (xy)' = x' + y' The Attempt at a Solution Converting from Sum of Products to Products of Sums requires the following...
  23. F

    What is the Integral of Riemann's Sum with Square Roots?

    Homework Statement Convert the Riemann's Sum to an integral: (1/50) * [(sqrt(1/50)) + (sqrt(2/50)) + (sqrt(3/50)) ... + (sqrt(50/50))] Homework Equations The Attempt at a Solution (1/50) times Integral (upper limit 1 and lower limit 0) of sqrt(x) dx Is my solution correct?
  24. M

    Understanding the Relationship Between One-Way ANOVA and Regression Models

    Homework Statement Assume a one way anova model: Y_{ij} =\my + \alpha_i + e_{ij} where e are independent, normal distributed with variance sigma and expectation = 0 Define: z_{ijl} = 1 if i = l, and 0 else Show that: Y_{ij} = \mu + \sum_{i=1}^I \alpha_l z_{ijl} + e_{ij} Homework...
  25. G

    HELP Sums of Random Variables problem: Statistics

    HELP!Sums of Random Variables problem: Statistics Homework Statement 3. Assume that Y = 3 X1+5 X2+4 X3+6 X4 and X1, X2, X3 and X4 are random variables that represent the dice rolls of a 6 sided, 8 sided, 10 sided and 12 sided dice, respectively. a. If all four dice rolls yield a 3, what...
  26. A

    Sums of Integer Powers - C(s) Convergence

    Hello all, Is there a closed form expression for the convergence of C(s) = \sum_{n=1}^{N} n^{s} Cheers, Adrian
  27. marcus

    State Sums and (Quantum) Geometry: Frank Hellmann's PhD thesis

    http://arxiv.org/abs/1102.1688 State Sums and Geometry Frank Hellmann PhD Thesis, 106 pages (Submitted on 8 Feb 2011) "In this thesis I review the definition of topological quantum field theories through state sums on triangulated manifolds. I describe the construction of state sum invariants of...
  28. H

    Algebraic Sums of Currents/EMFs in Junctions/Closed Loops

    1. What causes the algebraic sum of the currents (signs included) flowing into each of the four junctions NOT equal to zero? 2. What causes the algebraic sum of the EMFs in each closed loop NOT equal to the algebraic sum of all the IR drops in each loop?For the first question, I thought that...
  29. M

    Approximating integral using riemann sums

    Homework Statement f: [0,1] -> Reals, f(x) = 3-x2 P={0,1/2,1} Find lower and upper Riemann sums, and approximate the definate integral using them and find the corresponding approximation error. Homework Equations The Attempt at a Solution So I tried finding the upper Riemann...
  30. M

    Absolute value of riemann sums

    Homework Statement I'm trying to prove that Sp|f| - sp|f| \leq Spf - spf Where P is a partition of [a,b] and f is function that is riemann integrable. Homework Equations The Attempt at a Solution So I get to a point where M = supf(x) and m = inff(x) then |M|(b-a) - |m|(b-a)...
  31. A

    Left endpoint approximation & Riemann Sums (Sigma)

    1. The problem statement, all variables and givennown data 1)FInd the nth left endpoint approximation Ln for f(x) = 3x^2-2 on [0,2]. What is the limit as n approaches infinity Ln in this case? 2)Evaluate: \sum45i=5 (2i-5) Homework Equations Ln = \sumNj=1 f(cj)(xj-xj-1) The...
  32. I

    How to calculate the lattice sums of BCC?

    How do I calculate the lattice sums A12 and A6 for a BCC structure? I have calculated the following so far: A12 = 8(1/1)^12 + 6(1/root2)^12 + 12(1/2)^12 + 16(1/root5.5)^12 + 8(1/root6)^12 = 8.097. Have i made any mistakes? Are my nearest neighbour values correct? Please help! o:)
  33. R

    Prove that if f is integrable, so is |f| via upper and lower sums

    Note that this question is very context-sensitive, and comes from Taylor & Mann Advanced Calculus, 3rd Edition (if you have this book, the question is on Pg. 539). Homework Statement If f is integrable, so is the absolute-value function |f(x)|. Prove this by showing that if s, S refer to f...
  34. K

    Prove Sums of Cantor Sets in [0,2]

    I'm supposed to show that the sum C+C ={x+y,x,y in C}=[0,2] a) Show there exist x1,y1 in C1 for which x1+y1=s. Show in general for any arbitrary n in the naturals, we can always find xn, yn in Cn for which xn+yn=s. b) Keeping in mind that the sequences xn and yn do not necessarily converge...
  35. L

    Sums of Series: Find Sum with n=1 to ∞

    Homework Statement 2/(n+8)(n+6); find the sum of the series with n=1 to infinity. Homework Equations Hmm.. The Attempt at a Solution I tried to split this into a partial fraction and try to find the limit, but for some reason that didn't work for me. I'm confused as to approach...
  36. L

    Convergence of Infinite Series with Variable Terms?

    Homework Statement Sum from 0 to infinity of (2^n + 6^n)/(2^n6^n) Homework Equations No idea. The Attempt at a Solution I am completely dumbstruck on how to do this one. Could someone give me a hint on where to start? Thanks a lot!
  37. F

    Convert Expression to Sum of Prod & Prod of Sums

    Homework Statement Convert the following expression into sum of products and products of sums (AB+C)(B+C'D) Homework Equations Distributive Property The Attempt at a Solution for product of sums it would be (AB+C)(B+C'D) since it is already in this form. When calculating sum of...
  38. B

    On the sums of elements of uncountable sets

    I want to prove the following proposition: Given any uncountable set of real numbers S, there exists a countable sub-collection of numbers in S, whose sum is infinite. Please point me in the right direction.
  39. K

    Converting a boolean expression into simplest product of sums

    Homework Statement Given F = (A'+B')[ABD' + A'C + A'BD], simplify into a simplest product of sums.Homework Equations The Attempt at a Solution Multiplying through gives me A'BD + A'C. I have tried expanding, adding consensus terms, but I'm not sure how to take it from there. Additionally, I...
  40. jegues

    Combining Sums: Simplifying with Identical Exponents | Homework Example

    Homework Statement I'm having some confusion about combining sums. Our goal when combining these sums is to have the, (x-c)^{\text{whatever}} term to be the same in both sums. My confusion is better explained in an example. (see below) Homework Equations The Attempt at a...
  41. C

    Estimating area with finite sums

    Homework Statement Use the midpoint rule to estimate the area under the graph of f(x) = 7/x and above the graph of f(x) = 0 from [1,25] using two rectangles of equal width. Homework Equations N/A The Attempt at a Solution So first I found \Deltax by using (b-a) / n and got (25 - 1)...
  42. P

    Use Upper And Lower Sums To Evaluate An Integral

    Homework Statement The question is to use upper and lowers sums, Un and Ln, on a regular petition of the intervals to find the integral from 1 to 2 of f(x) = [[x]], where [[x]] is the greatest integer function.Homework Equations \Deltax = \frac{b-a}{n} The Attempt at a Solution \Deltax =...
  43. A

    Why Use Supremum Instead of Maximum in Riemann Sums?

    Let f be a Riemann integrable function defined on an interval [a,b], and let P = \{a = x_0 < x_1 < \ldots < x_n = b\} be a partition of [a,b]. I don't understand why the definition of (say) the upper Riemann sum of f associated with P is always given as U(f,P) = \sum_{i=1}^n M_i (x_i -...
  44. A

    Expected value of random sums with dependent variables

    Hi all, I have a question of computing the expectation of random sums. E(sim_{k=1}^N X_k) = E(N)E(X) if N and X_1, X_2,...are independent and X_k's are iid. Here both N and X_k's are r.vs. But the condition of N and X_1, X_2,...being independent is not true in many cases. How will...
  45. L

    Density Function for Sums of Random Variables

    Homework Statement Given the joint density, f(x,y), derive the probability density function for Z = X + Y and V = Y - X. Homework Equations f(x,y) = 2 for 0 < x < y < 1 f(x,y) = 0 otherwise. The Attempt at a Solution For Z = X + Y, I can derive the fact that, f_Z(z) =...
  46. S

    How do I combine series sums into one representation?

    Homework Statement I have these three series here \sum_{n=2}^{\infty} a_n \cdot z^n - \sum_{n=1}^{\infty} a_n \cdot z^{n+1} + \sum_{n=0}^{\infty} a_n \cdot z^{n+2} what I would like to do is to add the together as one sum representation The Attempt at a Solution From what I...
  47. Q

    Magnetic field between two wires (vector sums)

    Homework Statement Two parallel wires carry a current I and 2I in different directions. What is the magnetic field halfway between the two wires? Homework Equations Ampere's law Int (B dot dA) = permissivity x enclosed current The Attempt at a Solution Draw a circle around wire 1...
  48. S

    Estimating Sums of Alternating Series help

    Homework Statement [PLAIN]http://img696.imageshack.us/img696/3438/46981606.jpg Homework Equations The Attempt at a Solution in those squars, I am sure about everything that i did and i get it wrong.. the only thing i don't know is bn+1 how would i know if its =, < or > than...
  49. T

    Solve Physics, Chemistry & Math Problems - Grade 10

    HI guys, I don't know where should I post this but well here it is! Can anyone tell me where to find some practice problems on physics, chem and math online? (I hope they are free and interesting!) I am in grade ( class) 10 in school. And since these are my vacations I thought of doing...
  50. M

    Proof of Existence of z0 for Alternating Series of Real Numbers

    Homework Statement Let {a_n} be an alternating series of real numbers approaching zero. Prove there exist a z0 on the unit circle such that the power series \sum_{}^{} a_{n}z^{n} is uniformly convergent on the domain of z such that both |z|<=1 and |z-z0|>= delta, where delta>0 Homework...
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