# Talk:Summation

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## 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

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.

$\sum_{i=a/b}^{c/d} m = m\left(\frac{ac - bd}{ad}+1\right) \mbox{ iff } ad \ne 0$ $\sum_{i=a/b}^{c/d} m = \frac{c - a + 1}{d - a + 1} \mbox{ iff } d - a + 1 \ne 0$
$\sum_{i=a/b}^{c/d} m = m\left(\frac{ac-bd}{ad}+1\right) \or \frac{c-a+1}{d-a+1} \mbox{ both iff } a, d \ne 0 \and d-a \ne -1$

provided that $a, b \in \mathbb{Z} \and a, b \ge 0 \and a \ne b$ 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

### 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

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).

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.

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)

I agree, this article does a terrible job of explaining the concept in a way that is easy to understand.BenW (talk) 05:16, 6 May 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

Σ6k2 where k=2 is 22+32+42+52+62+72, but shouldn't it just be 22+32+42+52+62 (without the 72)?

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? Temperaltalk 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 6y1/2 while
• lower limit is x = 2y
• the general term is the square root thing

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

One extra pair of brackets $\sum_{x = 2y}^{6y^\frac {1}{2}} \sqrt[n]{x}$:
$\sum_{x = 2y}^{6y^\frac {1}{2}} \sqrt[n]{x}$
--Salix (talk): 22:31, 21 October 2008 (UTC)

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

## 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 (talkcontribs) 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

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 (talkcontribs) 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:
$\sum_{1 \le i \le n,\ i \text{ odd}} i$
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 nm + 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)
The change to which you refer was here. When you refer to "you", you do not seem to have any specific person in mind – we are all editors, so "we" would be more appropriate in this context.
The two formulae n(n+1)/2 and (n2+n)/2 are equivalent; it is a subjective choice as to which to use (or which is "better"). As it currently occurs in the article is probably more commonly used, and relates intuitively to a visual construction (stack rows of squares on top of each other, starting with n long working down to 1, duplicate the result, rotate and slot together to make a rectangle of area n × (n+1), the original sum being half this, i.e. n(n+1)/2. So this might be argued to be "better". — Quondum 11:45, 1 May 2012 (UTC)

Thanks! Yes intrigueing isn't it. I prefer the older version since it does not require a "visual" to grasp since it straight forward adds to the summed length(size) of the set on the left but you gave a good reason and helped us understand what its all about. Interesting that some empirical formulas have both a "left" and "right" or "odd" and "even" orientation to them. I will even use this little dialogue about this in my paper where I'm using it!

The Induction equation in question has changed many times in the article over the past year and I just needed to grasp the content a little better and ask some questions. Alls well that ends well.

Regards,

173.238.43.211 (talk) 12:54, 1 May 2012 (UTC) G2theF

## Less than 2 terms?!

Hello to Wikipedia Community,

This is my first ever Wikipedia talk post. Please excuse my stupidity!

I am a dumb, ignorant, old guy going back to school to learn math. So, I read this article and find the statement "It is possible to sum fewer than 2 numbers". Really? That seems, to me, anyway, to violate the logic of a definition of a sum; a sum means we are adding one value to another value (or more values). Why would we even bother to try to denote this impossible concept of a sum of one or zero values? That just does not make sense to me. Wouldn't we just instead say that the starting conditions "m = n" or "m > n" are just not allowed or undefined, in the same manner as we might say "any value divided by zero is undefined"? Please explain.

Cheers, Allan — Preceding unsigned comment added by Allan.w.macdonald (talkcontribs) 02:12, 7 September 2012 (UTC)

You deal with sums of a single number all the time. Every time you go to a store and buy a single item, there's a line on the receipt that says "total" (er, well, "subtotal") with just the price of that one item, right? So the sum of a single item is really not an impossible concept—it's a very useful concept, in fact. —Bkell (talk) 03:01, 7 September 2012 (UTC)
A sum of zero numbers is similar. Imagine that you have a bank account into which you make no deposits for a certain month. At the end of the month you will get a statement from the bank with a line that says something like, "Total deposits \$0.00." That's a sum of no numbers! —Bkell (talk) 03:04, 7 September 2012 (UTC)

## Sum of the first "n" cubes - even cubes - odd cubes (geometrical proofs)

The "IDEA"

— Preceding unsigned comment added by Ancora Luciano (talkcontribs) 18:53, 17 July 2013 (UTC)

New method for summing the first "n" cubes
Sum of the first "n" even cubes

— Preceding unsigned comment added by Ancora Luciano (talkcontribs) 19:11, 17 July 2013 (UTC)

Sum of the first "n" odd cubes

— Preceding unsigned comment added by Ancora Luciano (talkcontribs) 19:26, 17 July 2013 (UTC)

Starting from the basic idea described in the first animation, we introduced a new procedure to obtain formulas for summing the first "n" cubes, even cubes and odd cubes. This method, called "Successive Transformations Method", consists in an inductive handling of a geometric model, in order to obtain another equivalent which gives evidence of the searching formulas. — Preceding unsigned comment added by Ancora Luciano (talkcontribs) 19:47, 17 July 2013 (UTC)

See the animations.

Consider the final transformation that you see in the second animation, and we compare this figure with the square base parallelepiped that contains it. We expect that the ratio between these figures becomes 1/2 to infinity. Performing calculations with Excel, you see that this is true. In addition you encounter these other amazing results:

      n
lim (Σn n3)/(Σn n). n2 = 1/2
n→∞   1
n
lim (Σn n5)/(Σn n). n4 = 1/3
n→∞   1
n
lim (Σn n7)/(Σn n). n6 = 1/4
n→∞   1
n
lim (Σn n9)/(Σn n). n8 = 1/5
n→∞   1


which, by induction, can be generalized in a formula. Note that the inverse limit (Σ fom zero to n) diverges, reproducing the sequence of natural numbers. Note that the denominators of the results are the positions of the exponents in the numerator in the sequence of odd numbers. --Ancora Luciano (talk) 19:31, 17 July 2013 (UTC)

## First Summation Example

I have a question about the very first example

What is the period at the end of the expression? Sometimes its there, sometimes its not. Not a decimal point or a multiply symbol? So I agree with the others, find the article confusing - sorry

88.104.138.228 (talk) 09:57, 1 June 2013 (UTC)

Its actually just a full stop to end a sentence. Taking the first two expressions in the article:

Using this sigma notation the above summation is written as:

$\sum_{i=1}^{100}i.$

The value of this summation is 5050. It can be found without performing 99 additions, since it can be shown (for instance by mathematical induction) that

$\sum_{i=1}^ni = \frac{n(n+1)}{2}$

for all natural numbers n.

In the first there is a full stop as its the end of sentence, in the second the sentence continues so no full stop.--Salix (talk): 11:21, 1 June 2013 (UTC)

## Sum of the first "n" squares - even squares - odd squares (geometrical proofs)

The IDEA

— Preceding unsigned comment added by Ancora Luciano (talkcontribs) 16:05, 16 July 2013 (UTC)

New method for summing the first "n" squares
Sum of the first "n" even squares
Sum of the first "n" odd squares

— Preceding unsigned comment added by Ancora Luciano (talkcontribs) 13:10, 18 July 2013 (UTC)

Starting from the basic idea described in the first animation, we introduced a new procedure, called "Inscribed Pyramid Method", to obtain formulas for summing the first "n" squares, even squares and odd squares. The formulas are obtained by adding to the volume of the inscribed pyramid the external "coating" components.

--Ancora Luciano (talk) 13:17, 18 July 2013 (UTC)

## Minus one twelfth

FWIW - I don't know if my edit (based on a recent WP:Reliable Source) (recently reverted by User:DVdm) is entirely ok or not - but seems worth a discussion:

Interestingly, the summation of all natural numbers to infinity is "minus one-twelfth".< ref name="NYT-20140203">Overbye, Dennis (February 3, 2014). "In the End, It All Adds Up to –1/12". New York Times. Retrieved February 3, 2014.</ref>

$\sum_{n=1}^{\infty} n = - \frac {1}{12}$

ALSO - A relevant video (07:49) by Numberphile "proving" the notion is at the following => http://www.youtube.com/watch?v=w-I6XTVZXww - A related discussion is ongoing at Talk:Infinity#Minus one twelfth - Comments Welcome of course - in any case - Enjoy! :) Drbogdan (talk) 17:17, 4 February 2014 (UTC)

I have removed it again per wp:NOR. We have no source that says that it is a "procedure that can be justified in terms of Abel summation." And of course the New York Times is a newspaper, which is hardly a reliable source for mathematics. - DVdm (talk) 14:35, 6 February 2014 (UTC)
Well there are plenty of reliable sources which show that for particular methods of assigning values to infinite sums then -1/12 is an appropriate answer. See the 1 + 2 + 3 + 4 + ⋯ article. However I would say this is WP:UNDUE as in normal mathematics the series is divergent with an undefined answer. There may be a case to briefly mention zeta function regularization and Ramanujan summation two techniques which can be used to give values to divergent series.--Salix alba (talk): 16:11, 6 February 2014 (UTC)
It looks like user Tkuvho doesn't understand what original research means (no reference is provided), and is close to wp:3RR as well.
By all means go ahead and remove it per wp:NOR and wp:UNDUE, or turn it into something less silly. - DVdm (talk) 16:31, 6 February 2014 (UTC)
The reference I provided by Sondow specifically speaks of Abel summation. Are you implying that the Proceedings of the American Mathematical Society is unreliable like the New York Times? What's OR exactly? Tkuvho (talk) 16:43, 6 February 2014 (UTC)
Aw, I thought your edit summary referred to the Wikilink to Abel summation. That would have been OR. I hadn't noticed your addition of the Sondow ref. My apologies for having overlooked that and for underestimating your experience. IOU1. - DVdm (talk) 18:09, 6 February 2014 (UTC)
No problem :-) Tkuvho (talk) 18:18, 6 February 2014 (UTC)

The added material is wrong. The series $1+2+\cdots$ diverges properly to $+\infty$, and Abel summation (or any other kind of Banach limit) will not produce convergence. It is very misleading to say that the summation of all natural numbers is $-1/12$, since this is not true in the sense of summation discussed in the article. Appropriate context for this result needs to be given, and this article is not the place for discussion of zeta functional regularization. Sławomir Biały (talk) 01:24, 10 February 2014 (UTC)

The Sondow reference mentions this as an example of Abel summation. Zeta functional regularization is not needed. The appropriate context is Abel summation. Tkuvho (talk) 15:55, 10 February 2014 (UTC)
It's certainly wrong to say that $1+2+\cdots=-1/12$ is true by Abel summation of the series. Abel summation of a series of positive terms is equal to the supremum of the partial sums. So it's $+\infty$ in this case (as is the Cesaro sum, and any other kind of Banach limit you want to consider). Euler's trick was actually to look at the sum of a different series, one that was Abel summable, and then to relate it to the "value" of this series by a functional equation. So, while it is in a sense true that you can "prove" this using Abel summation at some point in the proof, it is very misleading as written. So much so, I think, that the article would not really benefit from a corrected version of it. Sławomir Biały (talk) 16:45, 10 February 2014 (UTC)
In fact, it could be argued that what Euler did actually was, or anticipated, zeta functional regularization, since he assumed what we nowadays recognize as the functional equation remained valid outside the region of convergence of the Dirichlet series expansion of the zeta function. That Abel summation is used at some point in the process seems rather peripheral in light of this. Sławomir Biały (talk) 17:02, 10 February 2014 (UTC)
Dennis Overbye looks like a sensationalist babbler with his article, and his New York Times publishers – as morons who creatively reinterpreted the minus sign as en dash. Certainly the Abel summation of 1 + 2 + 3 + 4 + ⋯ can’t give nothing but an infinity at z = 1 because the respective analytic function has a pole there. Drop it and forget. Incnis Mrsi (talk) 17:45, 10 February 2014 (UTC)
I haven't read either the NYT or Overbye but Sondow of the Proceedings of the American Mathematical Society mentions that Euler had three ways of evaluating this sum, one of them involving Abel summation. You may have noticed that Euler was fully a century before Riemann. Moreover Euler did not have the functional equation for the zeta function. Nonetheless Euler performed the necessary transformation of the series to (bypass the pole and) make it summable, and I see no reason not to mention this here. For details see here. Tkuvho (talk) 18:10, 10 February 2014 (UTC)
There is an article Ramanujan summation that needs an expert’s attention since 2008. Why experts on divergent series spend their expertise and creativity here, not where they are welcome? This article is about (simply) summation of which Ramanujan summation constitutes a minor subtopic. Incnis Mrsi (talk) 18:52, 10 February 2014 (UTC)
If you believe that the Abel sum of the series $1+2+\cdots$, then I invite you to try to prove this. Euler's Abel sum argument used the relationship between the zeta and eta functions. Both of these were known to Euler (ever hear of the Euler product?) The eta function happens to be Abel summable. But the argument is purely formal, since it applies the functional equation outside the domain where it is valid. Sławomir Biały (talk) 19:07, 10 February 2014 (UTC)
Some remarks:
• If the text under discussion would relevant to Wikipedia, it is clearly misplaced in the lead. At most, it could be placed in a section "Summation of divergent series", with a template {{main|Divergent series}}
• There are various methods for providing values to divergent series. This one, even if it is correct, is not the most important one (as far as I know, the most important one is the renormalization of the physicists, which consists in cutting the series at the least term, and had been mathematically justified at the end of 20th century by Malgrange, Ramis and others). Presenting Abel summation of divergent series alone would give WP:undue weight to this particular theory.
• Even if a clear and correct explanation could be provided, it would be WP:too technical for the lead of this article.
Therefore I fully agree with removing the contested paragraph. D.Lazard (talk) 19:14, 10 February 2014 (UTC)
Not to pile on, but there is an additional reason not to claim that Euler obtained −1/12, apart from any mathematical considerations. IMO, there is insufficient historical evidence. Euler definitely cared about evaluating the Riemann zeta function at positive integers. To that end, he evaluated the Dirichlet eta function at both positive and negative integers. (Of course, he would not have recognized our names for these functions!) And he did write that 1 − 2 + 3 − 4 + ⋯ = 1/4, essentially using Abel summation. It is plausible that Euler also might have evaluated the zeta function at negative integers, essentially using analytic continuation, but that is not supported by the primary sources. I know that Sondow claims that Euler calcuated ζ(−n) for n = 0, 1, 2, and 4. However, Sondow cites a huge list of references all at once for that claim, which is a red flag. I've looked up most of the references, and they don't support the conclusion. If, indeed, Euler ever wrote "−1/12" anywhere near the series 1 + 2 + 3 + 4 + ⋯, either in a letter or a publication, then it should be easy to cite a page number in a primary source. Failing that, all such claims should be interpreted as "Euler could have done this, but didn't."
All that said, I would be thrilled to be proven wrong! All it takes is one quotation from Euler himself. Does anyone have it? Melchoir (talk) 04:06, 12 February 2014 (UTC)
The question is, who is willing to invest time it would take to look for such a reference given the opposition to its inclusion here. This page seems an appropriate place to have a more advanced section on infinite summation, but there is no such concensus yet. I would be the first one to reject basing this on popular media reports, but just because the NYT writes about a mathematical subject is not yet sufficient reason to ban the said subject from Wikipedia. Tkuvho (talk) 12:11, 12 February 2014 (UTC)
Well, there is an article on 1 + 2 + 3 + 4 + ⋯. Since the Numberphile video was released, that article has had more page views per day than this one! The trend might change over time, but for now, it should provide some incentive to do the research and make sure that the popular article is correct.
About the best scope for this article, there's an argument to be made that infinite series should be better represented. I don't have a strong opinion, and I'm not sure that there's a clear consensus either way. You might want to try writing a section on infinite series. However, it probably shouldn't focus on the series 1 + 2 + 3 + 4 + ⋯. Melchoir (talk) 19:28, 12 February 2014 (UTC)