Talk:Summation

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[edit] 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

[edit] 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)

[edit] 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 \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

[edit] 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...

[edit] 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)

[edit] 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)

[edit] 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)

[edit] "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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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).

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