Talk:Summation

Growth of Harmonic row

It should be explained why the growth of the harmonic grow (sum 1/n) is approximated by Theta log(n) (no base given, our professor said base 2) but actually is ln(n) (base e). Is that of the same growth rate.

Infinite series

When b is replaced with the infinity (??) symbol, the sum is an infinite series. This has a countably infinite number of terms, and represents the limit of the sum of the first n terms, as n grows without bound.

This needs to be explained better. Vera Cruz

Theta

The lowercase theta function used on this page needs to be replaced by uppercase theta, as described at Big O notation. --Zero 06:30, 23 Dec 2003 (UTC)

Uncommon Bounds

See also Multiplication.

What if the bounds are fractions? For example the series:

$\sum _{i=1}^{n}2i-1=n^{2}$
$\left({\frac {a}{b}}\right)^{2}=\sum _{i=1}^{a/b}2i-1$
$\left({\frac {a}{b}}\right)^{2}={\frac {a^{2}}{b^{2}}}={\frac {\sum _{i=1}^{a}2i-1}{\sum _{i=1}^{b}2i-1}}$

Thus, it can be generalized that $\sum _{i=a/b}^{c/d}f(i)={\frac {\sum _{i=a}^{c}f(i)}{\sum _{i=b}^{d}f(i)}}$

Due to the commutative property of addition, $\sum _{i=a}^{b}f(i)=\sum _{i=b}^{a}f(i)$. Thus, with $b>a$, we iterate in reverse order (that is from the greater bound to the lower bound, or in reverse order) - for example:
$\sum _{i=1}^{3}i=1+2+3=6$
$\sum _{i=3}^{1}i=3+2+1=6$ (note the order)

What if the bounds are negative?

Also, $\sum _{i=-1}^{-3}i=-1+-2+-3=-1-2-3=-6$ and
$\sum _{i=1}^{3}-i=-1+-2+-3=-1-2-3=-6$ (note the sign at the bounds)

If $f(-i)=-f(i)\,\!$, then the generalization becomes
$\sum _{i=-a}^{-b}f(i)=\sum _{i=a}^{b}f(-i)=\sum _{i=a}^{b}-f(i)=-\sum _{i=a}^{b}f(i)$

What if the bounds are equal? In this case, the summation will yield the identity element for addition (that is zero or empty sum).

Thus, the generalizations are:

1. $\sum _{i=a/b}^{c/d}f(i)={\frac {\sum _{i=a}^{c}f(i)}{\sum _{i=b}^{d}f(i)}}$
2. $\sum _{i=a}^{b}f(i)=\sum _{i=b}^{a}f(i)$
3. $\sum _{i=-a}^{-b}f(i)=\sum _{i=a}^{b}f(-i)$
4. $\sum _{i=a}^{a}f(i)=0$
5. $\sum _{i=1}^{n}m=mn$ (see Multiplication)
6. $\sum _{i=a}^{b}m=m(b-a+1)$ (from the equation before this)
7. $\sum _{i=a/b}^{c/d}m$ is disputed because there are two possible definitions
   1. ${\frac {m(c-a+1)}{m(d-b+1)}}={\frac {c-a+1}{d-b+1}}$ according to #1
   2. $m\left({\frac {c}{d}}-{\frac {b}{a}}+1\right)=m\left({\frac {ac-bd+ad}{ad}}\right)$ according to #6

But we prefer both definitions, i.e.

$\sum _{i=a/b}^{c/d}m=m\left({\frac {ac-bd}{ad}}+1\right){\mbox{ iff }}ad\neq 0$ $\sum _{i=a/b}^{c/d}m={\frac {c-a+1}{d-a+1}}{\mbox{ iff }}d-a+1\neq 0$
$\sum _{i=a/b}^{c/d}m=m\left({\frac {ac-bd}{ad}}+1\right)\lor {\frac {c-a+1}{d-a+1}}{\mbox{ both iff }}a,d\neq 0\land d-a\neq -1$

provided that $a,b\in \mathbb {Z} \land a,b\geq 0\land a\neq b$ and that the ring is commutative over addition and that no quotient (divisor) is zero.

Critics and corrections are welcome. -- ErikDT

a question...

what is the result of: $\sum _{i=4}^{1}i$ ? if you look at the javascript code in the article page (http://en.wikipedia.org/wiki/Addition#Computerized_notation), i'd say the summation is zero. but this is not said in the definition of summation (http://en.wikipedia.org/wiki/Addition#Summation_notation)... looking at here (http://en.wikipedia.org/wiki/Summation) won't solve the problem... –

The sum of no numbers is 0 and the product of no numbers is 1. They are called the Additive identity and the Multiplicative identity. It is because 0 is the only number you can add to x without changing x, and 1 is the only number you can multiply x by without changing x. Lpetrazickis 16:18, 29 March 2007 (UTC)

The summation definition doesn't say what to do when the "upper bound" is smaller than the "lower bound" as in your example. Do you expect i to count backwards in that case, or is the JavaScript program correct? If summations should count backwards, this could be accomplished by swapping the upper and lower-bound variables if the low variable is

actually bigger. This would work because addition is commutative. It would be more efficient in those languages that support bitwise operations. Those that don't would need an extra variable. 216.23.105.5 11:21, 22 May 2007 (UTC)

The problem I've noticed with all the programming versions is that they can only represent the basic summation example. There needs to be an example of what two Sigmas next to each other means:

$\sum _{\ell }\sum _{\ell '}.$

There's no number to the right of either Sigma, so you don't have an argument to pass to Python's sum. In other areas of traditional math, two symbols of equal size next to each other implies multiplication. Furthermore, l is not initialized to a lower bound, and there is no upper bound. Does l therefore get a default initializer? Would you sum l from negative infinity to positive infinity and then multiply that with a similar summation of l-prime? Or is l actually an series^H^H^H^H^H^Harray, this fact being hidden by the ambiguous nature of traditional math notation? Or perhaps the second Sigma is the argument of the first Sigma. That raises the question of whether lazy evaluation is required to evaluate the above expression correctly (most programming languages evaluate arguments to functions before passing them, so they don't have to pass bits of machine code around. This strategy is known as eager evaluation. If this is how the above expression is evaluated, then the Sigma to the right would be evaluated into a scalar before the Sigma to the left is called with the result as an argument. Whether this returns infinity or something else depends on what the above form of Sigma does with the L.).

That the double-sigma expression above means the same thing as a single Sigma expression with L and L-prime separated by commas underneath it explains nothing. 216.23.105.5 11:21, 22 May 2007 (UTC)

Uh, yeah, I have the same gripe. To quote from the current article:

There are also ways to generalize the use of many sigma signs. For example,

$\sum _{\ell ,\ell '}$

is the same as

$\sum _{\ell }\sum _{\ell '}.$

I'm sure the article is correct -- but neither expression is explained further. Two undefined expressions have the same meaning. Great. What is symbolized by sigma with two terms underneath it separated by a comma? 24.211.245.220 (talk) —Preceding undated comment was added at 04:28, 3 October 2008 (UTC).

Am I stupid or just cannot calculate? I tried the $\sum _{i=1}i^{2}$ formula, and I get off-by-one result. ${\frac {10^{3}}{3}}+{\frac {10^{2}}{2}}+{\frac {10}{6}}=384$ but ${\frac {(10(10+1)(2*10+1))}{6}}=385$. What's this?

Atuomi (talk) —Preceding undated comment was added at 11:49, 10 October 2008 (UTC).

Off-topic definition

I've removed the text from the introduction

Summation can also be defined as cumulative action or effect; especially : the process by which a sequence of stimuli that are individually inadequate to produce a response are cumulatively able to induce a nerve impulse;

temporal summation

(noun) : sensory summation that involves the addition of single stimuli over a short period of time

spatial summation

(noun) : sensory summation that involves stimulation of several spatially separated neurons at the same time

If anyone wants to find a home for it, good luck! Melchoir 19:06, 29 November 2005 (UTC)

Hi does anyone know why the equation for the sum of x^3 is the equation for the natural numbers squared?

Split

I've split this article off from Addition. All of the above discussion has been moved from Talk:Addition, since it addresses the content now found at this article. Melchoir 02:12, 3 December 2005 (UTC)

Orphaned content

During the split, I didn't use the following content:

• One may also consider sums of infinitely many terms; these are called infinite series.

Notationally, we would replace n above by the infinity symbol (∞). The sum of such a series is defined as the limit of the sum of the first n terms, as n grows without bound. That is:

$\sum _{i=m}^{\infty }x_{i}:=\lim _{n\to \infty }\sum _{i=m}^{n}x_{i}.$

One can similarly replace m with negative infinity, and

$\sum _{i=-\infty }^{\infty }x_{i}:=\lim _{n\to \infty }\sum _{i=-n}^{m}x_{i}+\lim _{n\to \infty }\sum _{i=m+1}^{n}x_{i},$

for some integer m, provided both limits exist.

• In the case of repeated addition the augend is the first addend.

Melchoir 01:48, 3 December 2005 (UTC)

"Sigma notation" redirect

"Sigma notation" redirects to Addition. Shouldn't it redirect to Summation? Anybody looking for Sigma notation probably wants to know about summation, not just addition. -Leapfrog314 05:44, 3 December 2005 (UTC)

You're absolutely right, Leapfrog314; thanks for pointing it out. I have yet to crawl through Special:Whatlinkshere/Addition and Special:Whatlinkshere/Summation and fix links to point to the right places. You're welcome to do it yourself; if nobody does, I'll probably get around to it tomorrow. Melchoir 07:52, 3 December 2005 (UTC)

I've fixed all the redirects. Melchoir 10:22, 5 December 2005 (UTC)

Theta notation

I've removed the following:

"Basically, summating a function can be approximated by the Theta function of the antiderivative of the sequence being summated; this is true due to the integral test for convergance"

I'm sure there's a truth hiding in there, but

1. This article does not deal with infinite sums.
2. It's easy to construct pathological counterexamples, so the statement is wrong.

Does anyone know an analogous statement that is relevant and true? Melchoir 19:30, 27 December 2005 (UTC)

Okay, I've removed it again, this time reading:

"Because of the integral test for convergance, one can deduct that the sum of a sequence is equivelant to the theta function of the integral of the sequence."

Again, this statement is at best out of place and, as it is stated, wrong. If a sequence does converge, the growth rate of its partial sums is Order(constant), which is boring and doesn't apply to the sequences listed here. Melchoir 01:46, 28 December 2005 (UTC)

List formatting

I've reverted this change. The bullets are appropriate because the lists are lists, not prose with connecting text. The condensed spacing is appropriate because there is no danger of misunderstanding, and any extra space just pads out the article unnecessarily. Melchoir 08:00, 15 February 2006 (UTC)

Merge

I wanted a merge because the two articles are similar in topic. They both largely use ∑ (sigma) notation. You might as well describe what the symbol means before describing infinite series. Sr13 08:56, 22 November 2006 (UTC) I take my statement back....they should be seperate articles. Sr13 02:59, 23 November 2006 (UTC)

Overlap with Series (mathematics)

There is some overlap between the article on summation and the article on Series (mathematics). How about moving the section on "identities" from summation to Series (mathematics). Alternatively the page on Series (mathematics) could contain a link called finite series that that lead to the page on summation. The present situation is confusing, because it is difficult to guess that that the examples of finite series have been categorized as summation. —The preceding unsigned comment was added by 203.200.55.101 (talk) 06:06, 9 December 2006 (UTC).

I have added some links to a video Podcast

Hi everyone I am doing this following a suggestion by Eagle 101 I have added some links to a video podcast that I own. I think they are a nice addition to wikipedia please look at them and express you oppinion here ,judge if the links are really useful or not to wikipedia.

• It's all Greek to me! Sigma notation Video PodCast by http://www.isallaboutmath.com/
• Summation Telescoping Property Video PodCast by http://www.isallaboutmath.com/

If any of you think they are valuable to wikipedia then feel free to add them back in the external links. Regards SilentVoice 02:16, 22 January 2007 (UTC)

Hard to understand

It may be just me, but this entire article assumes a certain level of understanding of mathematics which I think is beyond the average reader. It's certainly a bit confusing for me (although I found the code examples made life easier, because I work in software development). Is this assumption deliberate? If so, can someone put in some useful links to sites that explain this in more layman's terms?

Thanks. Aidan 13:00, 16 April 2007 (UTC)

I agree with Aidan, I was completely lost trying to read it. Please make it clearer (I wanted to learn how the sigma works im math, not waste time.) Kris18 23:48, 6 May 2007 (UTC)

Hi, Aidan and Kris

I did not understand the article too. I think I understand about summation more than anybody in the world. To verify it look at at http://en.wikipedia.org/wiki/User:Ascoldcaves and for more details at www.oddmaths.info/indefinitesum. In the article there is a single summation formula to sum any analytical function with any summation boundaries. If you are a user of “Mathematica” there is at the bottom left of the page of the site a button “examples”. Opening the examples in “Mathematica” you can choose an analytical function at your choice and arbitrary complex boundaries and calculate your sum.

Ascold, Ascoldcaves (talk) 21:17, 4 March 2012 (UTC)

VBScript code needs fixing

I just fixed the Python code example, because it was doing $\sum _{i=m}^{n}i$ instead of $\sum _{i=m}^{n}x_{i}$. It looks to me like the VBScript code has the same problem, but I don't know enough of that language to fix it myself.

143.167.233.224 16:06, 7 May 2007 (UTC)

Actually, I just looked up the VBScript syntax and did it myself. Someone who actually knows the language might want to check that I got it right, though.

143.167.233.224 16:14, 7 May 2007 (UTC)

Excessive programming examples

Is it really necessary to show summation in 7 different computer programming languages? One, perhapas, but 7 seems excessive. --I80and 23:42, 8 May 2007 (UTC)

I agree that there are too many languages: There are are two examples from languages that have a sum function in their library (Python, Fortran/Matlab), and one language that is considered utterly obsolete, even harmful (Pascal). The JavaScript example is so similar to the C example that it adds nothing.

The use of more than one language is a necessity, however, because programmers of one language do not necessarily understand an example written in another (See above for a comment from somebody who doesn't understand Visual Basic). If we agree to use one programming language, some mathematician is going to push for a purely functional language like Haskell or OCaml (or worse yet, J), and then the programming examples will be just as inscrutable to non-mathematicians as the traditional notation. (Functional languages are those that have almost no flow-control keywords, so everything has to be done with recursion and anonymous functions called lambdas, which often need to be recursive themselves). 216.23.105.1 10:13, 22 May 2007 (UTC)

Removed JavaScript and Pascal examples

The text removed is below:

the following JavaScript program:

 sum = 0;
 for (i = m; i <= n; i++){
     sum += x[i];
 }

or the following Pascal program:

 Sum := 0;
 for i := m to n do
     Sum := Sum + x[i];

216.23.105.3 08:44, 23 May 2007 (UTC)

Error

Maybe I just don't understand this, but I think there's an error in the second equation of the notation section. It says

Σ⁶_k² where k=2 is 2²+3²+4²+5²+6²+7², but shouldn't it just be 2²+3²+4²+5²+6² (without the 7²)?

Please excuse the bad formatting.

208.114.164.79 15:54, 27 May 2007 (UTC)

Cyclic and symmetric sums

This article has no coverage of cyclic and symmetric sums whatsoever, which I find disturbing, especially as Wikipedia has no other articles on those subjects. Should I add them in in the sigma-notation section? Temperal^{talk and matrix?} 03:20, 11 December 2007 (UTC)

The phrases "cyclic sum" and "symmetric sum" do get occasional use, and these concepts should probably be treated on Wikipedia somehow.

However, I don't know how much use the sigma notations \sum_{cyc} and \sum_{sym} get, outside of books with the phrase "USA And International Mathematical Olympiads" in their title. If it's a notation that's just used by a few authors, I don't think it's notable enough to appear here. Melchoir (talk) 03:17, 29 January 2008 (UTC)

Name change

This article's name should be series, the real name for this sort of thing. Spaz man 12:22, 22 January 2007 (UTC)

We already have Series (mathematics). This article could probably use a better name, but that one's unfortunately taken. Melchoir (talk) 03:22, 29 January 2008 (UTC)

question

I don't know if I fully understand summation so could someone tell me if this is correct? $\sum _{k=8}^{n}k^{2}=8^{2}+9^{2}+10^{2}+...(n-1)^{2}+n^{2}$ Zrs 12 (talk) 02:18, 29 January 2008 (UTC)

Yes, but please try Wikipedia:Reference desk/Mathematics. Melchoir (talk) 03:22, 29 January 2008 (UTC)

How to you sum a negative power?

What's the solution for this?: $\sum _{i=1}^{n}i^{-1}$

I presume your question is how a negative exponent is interpreted. --69.91.95.139 (talk) 20:31, 6 April 2008 (UTC)

a good reference

sum of the 1st n integers, each raised to power p can be given as:

S<n^p>= sum (k=1 to n) [( n+1-k) .sum(m=1 to p) pCm.k^(p-m)(-1^(m+1))] —Preceding unsigned comment added by 220.227.207.194 (talk) 10:38, 23 June 2008 (UTC)

Can somebody help with this

How do you write the sigma notation so

• upper limit is 6y^1/2 while
• lower limit is x = 2y
• the general term is the square root thing

$\sum _{x=2y}^{6}y^{\frac {1}{2}}{\sqrt[{n}]{x}}$. --Ramu50 (talk) 20:55, 21 October 2008 (UTC)

One extra pair of brackets <math>\sum_{x = 2y}^{6y^\frac {1}{2}} \sqrt[n]{x}</math>:

$\sum _{x=2y}^{6y^{\frac {1}{2}}}{\sqrt[{n}]{x}}$

--Salix (talk): 22:31, 21 October 2008 (UTC)

Thanks --Ramu50 (talk) 21:44, 23 October 2008 (UTC)

C-Class

This article isn't great, or extremely descriptive. But it seems to fit the guidelines of a C-class article. Yes? Wikijjc (talk) 18:47, 31 December 2008 (UTC)

Summation (sigma) "operator" binding

The article lacks information on how the summation (sigma) "operator" binds.

We see $\sum xy$

but $\sum (x+y)$ which means that $\sum x+y$ means something different.

What are the rules for when the parentheses are needed (or else it would mean something different) and when they can be omitted (it means the same in both cases)? —Preceding unsigned comment added by IrekReklama (talk • contribs) 18:11, 28 January 2010 (UTC)

Sums

Times tables

1x1=1

2x2=4

3x3=9

4x4=16

5x5=25

6x6=36

7x7=49

8x8=66

9x9=81

10x10=100

adding

1+1=2

2+2=4

3+3=6

4+4=8

5+5=10

6+6=12

7+7=14

8+8=16

9+9=18

10+10=20

11+11=22

12+12=24

13+13=26

14+14=28

15+15=30

16+16=32

17+17=34

18+18=36

19+19=38

20+20=40

See You Again —Preceding unsigned comment added by Coolconnor2010 (talk • contribs) 15:10, 16 February 2010 (UTC)

"useful" identity?

$\sum _{n=s}^{j}f(n)+\sum _{n=j+1}^{t}f(n)=\sum _{n=s}^{t}f(n)$

Should be seen as intuitive as it just continues the summation (and its notation isn't even vague in suggesting otherwise). Does it have some importance marking its inclusion here, because in all my mathematical studies I haven't come across this property being noted once. 124.171.169.189 (talk) 20:26, 11 March 2010 (UTC)

Hi. Yes, it really is obvious. However, this property (as well as those before) is usually listed as one of the basic rules of manipulation with sums, let's say "axioms of summation", for formal reasons. Besides that, these equations could also help someone who has difficulties with sums. Best wishes, --Tomaxer (talk) 00:42, 12 March 2010 (UTC)

Incrementation other than 1?

Is there a way to define an incrementation/step other than 1 ("1,2,3,4,5..."), such as 2 ("1,3,5,7,9...")? In a "For-Next" instruction it is defined as "For TN = 1 To UT Step 2". Is there an equivalent modification for sigma notation? I realize you could just define the term as "2TN-1", but I would think that there would be a way to incorporate it into TN, something like

$\sum _{TN=1[+2]}^{UT}Q_{TN}\quad {\mbox{or}}\quad \sum _{{\overset {TN}{(+2)}}{}^{{}^{{}^{{}^{=1}}}}}^{UT}Q_{TN}\;?{}_{\color {white}.}\,\!$

 ~Kaimbridge~ (talk) 12:13, 9 April 2010 (UTC)

There is no standard notation for that. However, I have seen people write things like:

$\sum _{1\leq i\leq n,\ i{\text{ odd}}}i$

which does the job. (Normally I'd put a line break instead of a comma, but \substack isn't supported here. Grr.) Ozob (talk) 23:16, 9 April 2010 (UTC)

Empty summation: Zero or empty set

There is no reference given for the statement that empty summation evlaluates to zero. In an additive group (say), the empty set is the absorbing and idempotent element of Minkowski addition; this suggests that some care is needed, and that a reference would be useful. Thanks, Kiefer.Wolfowitz (talk) 15:07, 2 July 2010 (UTC)

Sum of 1, off by 1?

This seems like a trivial thing, but seems to be incorrect in the listing.

The sum of 1 from m to n is n - m + 1, not n - m. A trivial example, sum 1 from 1 to 10, the answer is not (10 - 1 = 9), it is (10 - 1 + 1 = 10). — Preceding unsigned comment added by 68.13.40.239 (talk) 04:46, 10 September 2011 (UTC)

Right; this is corrected now. As a general rule summation expressions would be simpler if the upper limit in a summation would have been the first value not to include rather than the last value to include, but it's too late to change that convention. Note that n + 1 occurs more frequently in the expressions of the section in question than n (and this is why I wrote n + 1 − m rather than n − m + 1). Marc van Leeuwen (talk) 08:44, 10 September 2011 (UTC)

Summation

If someone want to know how to sum one can look at www.oddmaths.info/indefinitesum. At the bottom left at the page of the site there is the button “examples”. One can choose any analytical function and arbitrary complex boundaries and sum the function. Before doing it it is useful to read “Instruction”.

If one want just to look at the summation formula one can look at at http://en.wikipedia.org/wiki/User:Ascoldcaves Ascoldcaves (talk) 22:36, 4 March 2012 (UTC)

I have found examples of the Induction example you use in the article on other articles BUT you had before a different example of an Induction equation that made more sense .... It was n squared + n /2 not what it is now which is n(n +1) / 2. The earlier version worked logically ( for example for any length string you add all the digits (summed) ex. for 5 the sum is 15 ... so 5 squared + 5 / 2 = 15 ... or ex. for 7 the sum is 28 ... so 7 squared + 7 / 2 = 28.

You had the better Induction example before ... why was it changed?

173.238.43.211 (talk) 08:13, 1 May 2012 (UTC)