What is Sigma notation: Definition and 60 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




{\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,







{\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. kai sinclair

    Efficient Calculation of Final Magnitude and Angle in Vector Addition Method

    okay so I'm a Electrician I've found short method of calculating the final magnitude of a system (Lₜ), this relies mainly on Eucliud's axiom of angles within parrallel lines and is this ∑(cos(θₙ-θₜ)⋅Lₙ)=Lₜ where Lₙ and θₙ are the initail manignitude and angles respectively, and θₜ is the final...
  2. U

    MHB Precalculus: What is the value of this sigma notation?

    Hi, I'm currently a Grade 11 student and I need help for this question (Precalculus): If $\sum\limits_{i=1}^{50} f(i)=90$ and $\sum\limits_{i=30}^{50} g(i)=60$, what is the value of $\sum\limits_{i=1}^{50} (7 g(i)-f(i)+12)/(2)$? P.S. To those who could answer this, it would be a great help...
  3. H

    Proof of a dot product using sigma notation

    Mentor note: Moved from a technical section, so is missing the homework template. Hi, I'm always not sure how to prove something in math and I'm wondering if this is enough. ##\vec r \cdot (\vec u + \vec v) ## ##\vec u + \vec v = (u_1+v_1, u_2+v_2,u_3+v_3) = \vec s## ##\vec r \cdot (\vec u +...
  4. Zack K

    Calculating area using sigma notation

    Homework Statement The question involves using sigma notation of Riemann sums to find the area under the graph of ##x^2+\sqrt {1+2x}##. I managed to calculate most of the values and I have ##16+\frac 8 3 + \Sigma {\frac 2 n \sqrt {9 + \frac {4i} n}}## Homework Equations [/B] ##\Sigma i= \frac...
  5. C

    MHB Math induction with sigma notation

    Prove by math induction that n sigma 3i + 1 = n/2 (3n + 5) i = n
  6. N

    Mathematical Induction with Sigma notation

    Prove by mathematical induction that n sigma r^3 = n^2(n+1)^2/4 r = 1 so far I have 1 sigma r^3 = 1^2(1+1)^2/2 r=1 1 = 1(4)/2 1 = 4/2 1 = 2 I'm not sure what to do after this for the k+1 case.
  7. RoboNerd

    I Question about simplifying Sigma notation

    Hello everyone! I have this expression which I have to simplify: ∑i=1log(n) 1(log(n) - i) And my book apparently simplifies it to being log(n). I am struggling to figure out why this is the case. Could anyone help? Thanks!
  8. H

    How to solve this addition?

    Homework Statement Homework EquationsThe Attempt at a Solution I tried to rationalize the fractions by multiplied it by $$\sqrt{n + 1} - \sqrt{n} $$ it will be sum of $$ \frac {\sqrt{n + 1} - \sqrt{n}} {2n +1} $$ also tried to see the arithmetic structure $$ 1 + \frac{1}{\sqrt{2} +...
  9. bluejay27

    I Representing sums as sigma notation?

    Hi is there a way or algorithm to find the sigma notation of sums in which the sums do not have an apparent general form?
  10. L

    Sigma notation and calculator help(casio fx9860g; ti-30xpro)

    Homework Statement A) use sigma notation and calculate sum as far as you wish (set your own endpoint)(1/2) + (1/4) + (1/8) + (1/16)...B) this is a personal problem because I want to know how to calculate the sums with e.g. sigma notation, with a calculator. Never having done this before...
  11. A

    Sigma notation question

    Homework Statement http://postimage.org/][/PLAIN] [Broken] image url Homework Equations N/A The Attempt at a Solution I have to solve a different question using a similar method to this one but I cannot figure out how they got the sigma notation into the last equation format. I tried writing...
  12. W

    Express a series in sigma notation

    Homework Statement I've been given the series 3,4,6,10,18... and asked to express as the ∑ar, between r=0 and r = infinity. Homework EquationsThe Attempt at a Solution Well, I can see a pattern! The difference between terms doubles every time. I'm having difficulty expressing this...
  13. P

    MHB Polynomial including Sigma Notation

    Hello everyone! I have this polynomial: $p(x) =$ 1 + \sum_{k=1}^{13}\frac{(-1)^k}{k^2}x^k - I'm supposed to show that this polynomial must have at least one positive real root. - I'm supposed to show that this polynomial has no negative real roots. - And I'm supposed to show that if $z$ is...
  14. J

    Sigma Notation and Product Notation

    Homework Statement Notice that ln [∏(k=1)^n a^k] = Ʃ_(k=1)^n * ln (a_k) I couldn't get the LaTeX right on this ^ But k=1 is below the product sign, and n is above. And (a^k) is the formula. From this, as well as some calculus, calculate that: lim as n->∞ ∏_(k=1)^n e^\frac{k^2}{n^3} For...
  15. Lebombo

    Can this be written in sigma notation: a + ar + ar^2 + + ar^n-1?

    Homework Statement a_{1}r^{1} + a_{1}r^{2} + a_{1}r^{3} +... + a_{1}r^{n-3} + a_{1}r^{n-2} + a_{1}r^{n-1} = \sum_{n=1}^{?} a_{1}r^{n-1} What value would replace the "?" 2. The attempt at a solution EDIT: I edited this post after receiving your reply Cepheid. My gut would say...
  16. MarkFL

    MHB Serina's questions at Yahoo Answers regarding sigma notation

    Here are the questions: Here is a link to the questions: How do you write the summation notation for: 4 - 24 + 144 - 864 + ... ? - Yahoo! Answers I have posted a link there so the OP can find my response.
  17. H

    How to write taylor series in sigma notation

    Homework Statement My Calc II final is tomorrow, and although we never learned it, it's on the review. So I have a few examples. Some I can figure out, some I cant. Examples: f(x)=sinh(x), f(x)=ln(x+1) with x0=0, f(x)=sin(x) with x0=0, f(x)=1/(x-1) with x0=4 The only one of those that I was...
  18. 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...
  19. H

    Why can't the indefinite integral be written with sigma notation?

    Hi, I've been wondering this since I started learning integration. I get that ∫ is basically an elongated S for "sum", because that is what it is basically doing. But then Ʃ does the same thing as well. If I'm understanding the difference, it is that Ʃ increments by finite measures, whereas ∫...
  20. B

    Matrix Multiplication and sigma notation

    Hello, I have read several different sources on this very topic, and the one thing that confused a little was defining it using sigma notation. Could some please explain to be what it means?
  21. S

    Prove of sigma notation

    Homework Statement Prove that: \sum_{n=1}^{14} 10n = \sum_{n=1}^{7} (20n+70) Homework Equations properties of sigma notation The Attempt at a Solution I know several properties of sigma notation but none that I know can be used to prove this. I don't know how to change 10 n to 20...
  22. rustynail

    Help with capital sigma notation please.

    I was playing a bit with the Riemann Zeta function, and have been struggling with some notation problems. The function is defined as follows \zeta (s) = \sum_{n=1}^{\infty} \frac{1}{n^s} where s \in \mathbb{C} we know that n^s = exp(s\;ln\;n) so I can write \zeta (s) =...
  23. K

    How to interpret this Sigma notation.

    Homework Statement Taken from Wikipedia's Palindromic number entry, under formal definition header: Although palindromic numbers are most often considered in the decimal system, the concept of palindromicity can be applied to the natural numbers in any numeral system. Consider a number n >...
  24. Y

    Simplifying Sigma Notation

    Homework Statement I'm actually asked to calculate the area under the curve 5x + x2 over the interval [0,1] using Riemann Sums. I found the formula for the Riemann sum over the interval, it being the following: \sum^{n}_{k=1} (\frac{5k}{n}+\frac{k^{2}}{n^{2}})(\frac{1}{n}) However, I...
  25. V

    Sigma Notation help requested

    Homework Statement "Express the following sum in sigma notation: 1 + (2/3) + (3/5) + (4/7) + (5/9)" Homework Equations The Attempt at a Solution I've figured out what they all have in common (1+2=3, 2+3=5, 3+4=7, 4+5=9) but I've been searching through the book and on the...
  26. J

    Proof By induction Sigma notation Please help

    Homework Statement 2n-1 Sigma (3i+1) = n(6n-1) i=0 prove for all positive n Homework Equations The Attempt at a Solution It holds true for n=1 5=5 then P: m+1 2m+1 Sigma(3i+1) = (m+1)(6(m+1)-1) or 6m^2 + 11m + 5 i=0 then 2m+1 Sigma(3i+1) = m(6m-1) +...
  27. I

    Summing Series (Sigma Notation)

    Homework Statement a) nƩr2(r-1)r=1 Homework Equations Using the summation series formulae... The Attempt at a Solution So far I have got: r2(r-1) = r3-r2 Ʃr3 = \frac{1}{4}n2 (n+1)2 Ʃr2 = \frac{1}{6}n(n+1)(2n+1) Therefore, Ʃr3-r2 = \frac{1}{4}n2 (n+1)2 -...
  28. R

    Help with sigma notation where the top value is x?

    Homework Statement In each case, x is an integer between -6 and 6 inclusive. Homework Equations x Σ 2i =12 i=1 The Attempt at a Solution 2x1 = 12 + 2x2 = 12 +...+2x(x) = 12
  29. B

    Sigma Notation: Definition and Examples

    What does this mean? (see attachment) "Where b is a block defined by the contiguity condition c that may exist between elements of s, and n is the number of elements in that block" I know is not possible to get a solution without the actual function, but how does this reads?
  30. Saitama

    Converting Sigma notation

    Converting Sigma notation... Homework Statement (Not a homework question) Hi! I have been encountering problems in Binomial Theorem which includes converting the sigma notation. Like \sum_{k=0}^n \frac{n!}{(n-k)!k!} a^kb^{n-k}=(a+b)^n I got many questions in my exam of this type with four...
  31. M

    Sigma Notation: Learn How to Evaluate Expressions

    Express this in sigma notation? 3^3 - 3^4 + 3^5 - ... - 3^100 Evaluate these two sigmas? n ∑ (i-2)^2 i =1 n ∑ (4-i^2) i =1 I don't really understand sigma notation so I'm really interested in the process and explanation of how to do it. Any help would be greatly appreciated!
  32. J

    Sigma Notation for Λ1 + Λ2 + Λ3 +...+ Λn = 1

    How do I notate this: \Lambda_{1}+\Lambda_{2}+\Lambda_{3}+\ldots+\Lambda_{n}=1 using sigma notation?
  33. A

    Simplfication of a sigma notation

    Homework Statement 1/1(4) + 1/4(7) + 1/7(10)+...+ 1/(3n-2)(3n+1) and the sigma notation is pretty obvious Homework Equations nothing really.. The Attempt at a Solution I can see an obvious pattern its n(n+3) but n cannot be 2,3,5,6 etc.. and the second digit in the first term...
  34. M

    Evaluation of Sigma Notation

    The problem posed is: Evaluate \sum_{i=1}^{n} (2i + 2^i) I know that I can break the summation down to this: 2\sum_{i=1}^{n} i\, +\, 2\sum_{i=1}^{n}1^i and then after using some Fundamental Theorems... =2\Bigg(\frac{n(n+1)}{2}\Bigg) + 2^n I can't seem to get it to look like my answer...
  35. F

    HELP how to plug this into the calculator sigma notation?

    how would i plug this into the calculator? to find the sum? step by step process? thankss! 4 ∑ 2 cos (π/k) k = 1 i tried this site http://mathbits.com/mathbits/tisection/Algebra2/summation.htm [Broken] and i got a different answer...so i just need help with the set up on a T.I calc.
  36. D

    What is the meaning of the sigma notation in this matrix A?

    I've been clawing at my mind for a while regarding this, I pray somebody can help me. I was implementing a method from a computer science paper when I came across this: [PLAIN]http://www.mattkent.eu/challenge.png [Broken] Bearing in mind that x is a vector, epsilon is a vector of...
  37. F

    Write the sum without sigma notation and evaluate it?

    4 ∑ 2cos (π/k) k = 1 i have the answer..but all i need it the steps on how to figure it out! thanks!
  38. P

    Expanded Form of Sigma Notation Problem

    Write the sum in expanded form. I don't know where to start with...
  39. R

    How to express vector in sigma notation

    A vector A, can be expressed in sigma notation as \sum Ai where i runs from 1 to 3, i.e. A1 for x coordinate, A2 for y coordinate and A3 for z coordinate. I wonder how to express vector A in polar form using sigma notation. Could anyone share their knowledge to me?
  40. B

    Series to Sigma Notation

    1. Write each series using sigma notation. 2. The series is: 2 + 5 + 10 + 17 + 26 + 37 3. I know that above the sigma, the number is 6 because there are only six terms in the series. The trouble I seem to have with this is the fact that the series is neither arithmetic or geometric. It...
  41. R

    Sigma notation and parentheses question

    I have a question about sigma notation and parentheses. Does \sum x[k] + x[n] mean \sum( x[k] + x[n]) or \left(\sum x[k] \right)+ x[n] I find it a personal annoyance for things to be written with parentheses even if there are rules about it. Including parentheses would just reduce...
  42. S

    Use sigma notation for expression

    1. The problem statement, all variables and given/nknown data Find a formula for the expression. Use sigma notation. (x + 1/y)n The Attempt at a Solution Not sure how to do this we have only dealt with sigma notation involving only n as a variable
  43. H

    Confused on sigma notation

    I understand how to solve sigma notation problems where the index variable is equal to 1, but how would I solve a problem like this?? n \Sigma (i2-3) i=3 If i was equal to 1 I would be able to solve this, but I'm not sure what to do since it is equal to 3. Thanks!
  44. U

    Sigma notation and telescopic problem

    Homework Statement Evaluate the telescoping sum. a) Sigma i=1 to n [i^4 - (i - 1)^4] Homework Equations sigma i=1 to n [f(1) - f(i+1)] The Attempt at a Solution so i plug in for the eqns. and get 1^4 - n^4 the correct answer should be n^4 but don't know how to get that.
  45. gn0m0n

    Center of mass prob. with sigma notation, vectors

    Homework Statement Prove the magnitude R of the position vector \vec{R} for the center of mass from an arbitrary point of origin is given by the equation M^{2}R^{2} = M\sum{m_{i}r^{2}_{i} - \frac{1}{2}\sum{m_{i}m_{j}r_{ij}^{2} Homework Equations \vec{R} = \frac{1}{M}...
  46. B

    Proving Sigma Notation Equals Zero

    Homework Statement Show: \sum_{i=1}^n (x_i - \overline{x}) = 0 Homework Equations Sigma notation The Attempt at a Solution \sum_{i=1}^n x_i - \sum_{i=1}^n \overline{x} = \sum_{i=1}^n x_i - \frac{1}{n}\sum_{i=1}^n \sum_{i=1}^n x_i = 0 \sum_{i=1}^n x_i = \frac{1}{n}\sum_{i=1}^n...
  47. L

    Sigma Notation Question/Trig Identity

    [SOLVED] Sigma Notation Question/Trig Identity I posted this elsewhere but I think I put it in the wrong place so I'm going to post my question again here. Basically I have to deduce the second formula from the first. Both equations are the same except for the top of the right side...
  48. L

    Learn How to Solve Sigma Notation Problems | Step-by-Step Guide

    I have no idea how to use the forum equation code so I just attached the question as a word document. I have no idea how to start this one. Any help is appreciated. Edit: Ok I just took a screenshot and uploaded it as an image.
  49. S

    Sigma notation of a series.

    [SOLVED] Sigma notation of a series. I have the formula 1+2+3+...+n = (n^2+n+1)/2, and I thinkthat this is the formula for the sum of a series. I need to write this thing in sigma notation, and then prove it by induction. I'm usually good and proving things by induction, but I can't...