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summation

Defines: 
running
Keywords: 
sum, sigma, notation, evaluating, progression, sequence
Synonym: 
sum, summing, sigma notation, $\sum$
Type of Math Object: 
Topic
Major Section: 
Reference

Mathematics Subject Classification

11B25 no label found00A05 no label found

Comments

Which is the value of the sum

SIGMA_{i = 1}^4 of 2k+1,

is it 21 or 24? I have always (in 44 years) seen only such a practice, that the summing operator, similarly as the deriving and integral operators, affects only the first term (or addend) on its right side. This implies e.g. that SIGMA x + SIGMA y does not mean double summing
SIGMA (a + SIGMA b).

Which is the value of the sum

SIGMA_{i = 1}^4 of 2i+1,

is it 21 or 24? I have always (in 44 years) seen only such a practice, that the summing operator, similarly as the deriving and integral operators, affects only the first term (or addend) on its right side. This implies e.g. that SIGMA x + SIGMA y does not mean double summing
SIGMA (a + SIGMA b).

Hi pahio,

If I'm reading your notation correctly you want to calculate:
SIGMA{i=1..4} (2*i+1)? (i.e. (2*1+1) + (2*2+1) + (2*3+1) + (2*4+1))?

Basically we have two properties of sigma we can use:
SIGMA(a+b) = SIGMA (a) + SIGMA (b) [where the range remains the same]
and SIGMA(c*a) = c * SIGMA (a) where c is a constant

so SIGMA{i=1..4} (2*i+1) = 2*SIGMA{i=1..4} (i) + SIGMA {i=1..4} (1)
= 2*SIGMA{i=1..4} (i) + 4
= 2 * 10 + 4 [using SIGMA{i=1..n} (i) = (n*(n+1)/2)]
= 24.

So no, there is no SIGMA double summing as you had presented:
SIGMA (a + SIGMA b).

Karli

and...
http://planetmath.org/?op=getobj&from=objects&id=6361

also gives these and other properties of SIGMA

I would have said it is

(2*1+1) + (2*2+1) + (2*3+1) + (2*4+1) = 3+5+7+9 = 24.

I agree that the SIGMA stops being active if it encounters
another SIGMA, but I think that it continues to be active after a +,
by default.

But of course if one has the option, one can and should use parentheses
to be completely clear.

I've recently had to endure some pain due to EXPECTATION X^2 -- no parens!

This reminds me of a saying
"write not to be understood, but so that you cannot be
misunderstood"

To make things precise, there should be parenthesis around
2i+1. Or if you really want to add one to the sum, then write
1 + \sum 2i

For simple sums, I think the below expressions are
unambigious:

1) b + \sum a_i

2) \sum a_i + b_i
(meaning \sum (a_i + b_i). This is implied by the
index in b_i)

3) \sum a_i + \sum b_i = (\sum a_i) + (\sum b_i)
(and not \sum (a_i + \sum b_i))

However, for any more comlpicated sums, I think that parentheses
should be used as much as necessary so that the sum is
readable. Clarity should be the aim.

I usually consider the "big operators" (\sum, \bigcup, etc) having lower precedence than "small operators" (+, \cup, etc)
so SIGMA 2k + 1 I take it as all parent post did:
(2*1+1)+(2*2+1) + (2*3+1) etc
and as matte points, that's specially true when the summnds all involve the summation index
\sum a_k + b_k
makes no sense expanding as (\sum ak) + bk

HOWEVER in case of doubt, and complex sums (well the example was a bit simple so it was understood out from context), it's much better to use parentheses

NOtice also that (on the top post), integral has the differential delimiting its scope, so it's not as good as analogy

f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

Matte is right, and I agree with him [her?] -- one should strive for precision and unambiguity in mathematical expressions. Therefore one should use parentheses after a sum sign when one means to sum an expression with more than one "terms"; dropping the parentheses is thus in principle worse than using them (so I hope that here in PM we should not set bad examples in this thing; cf. "summation").
Another motive: WHY should the summing operator behave in different way as other operators, e.g. deriving operators (d/dx, D etc.), nabla, integral operators? These all require parentheses for affecting sums and differences -- d/dx sin2x + cos3x is not same d/dx (sin2x + cos3x) in spite of the common variable x.

Jussi

Don't get mad at me, but if I wanted to mean
(\sum 2k) + 1 I would write it in THAT particular way or as matte sugests
1+\sum 2k

and I think it's customary to understand that a summation "closes" revious summations
so \sum 2k + \sum 1 is not \sum (2k + \sum 1)

I ALSO said that when in doubt and/or complicated expressions, parenthesis should be used for the sake of clarity.

But let's not be too pedantic, and I don't think my entry sets a bad example or it's confusing (unless you WANT to be confused)

Notice all the first posters (stux and jac) that answered the original question said it was 24 so that means it's how a given mathematician would understand it out of context.
And I guess the particular place that bothers you was understood the proper way by everyone who looked at it.

HOWEVER, I'll grant you write access to ALL of my entries so you can fix whatever you take in consideration (so you don't think I'm stubborn or maliciously trying to confuse readers)
f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

We all have seen that the use of parentheses is a matter of taste -- e.g. drini and I evidently have the opposite tastes =o) I know that its always dangerous to dispute of such matters and hope that I have not made drini angry.
I have wanted to bring up my pedantic point of view, which is based on the similar behaving of operators. The cause was that I have in PM seen in two places the practice without parentheses (it shocked me since I have not before seen such, or do not remember such).
All in all, it is good that we discuss also the use of parentheses -- it belongs to mathematics.

Jussi

An alternative to paretheses is a clearly stated convention.
If we agree that ^ has greater precidence than +, for example,
there is no need to parethesize 2+2^3. Otherwise, if we don't
agree, then there is potential for confusion.

Aaron has an entry called OrderOfOperations, I think this would
be the correct place to take up discussion of the conventions
used for SIGMA (and differentiation, etc.) on PlanetMath.

Hi guys,

I wouldn't be surprised if there is already a clearly established convention for handling order of operations when the SIGMA or \sum operation is involved. However after a quick search I have not found reference to this topic.
However I do beleive, as drini stated, that the established convention is to give \sum a lower operator precedence than '+'. That means in the same way 2^3+4=(2^3)+4 differentiates 2^(3+4) (as you pointed out jac), \sum a + c should necessarily be understood as (\sum a) + c or c + (\sum a) as an clearer alternate notation drini suggested.
That said, it is my opinion, having \sum a_i + b_i (although understood to be part of \sum in context,) should be grouped by a parenthesis as \sum (a_i + b_i) for clarity.
An example of this can be seen in Wikipedia's entry for SERIES where they state an example as \sum (b_n-b_[n+1]) where the parenthesis is included for clarity.
The link to this is:
http://en.wikipedia.org/wiki/Series_%28mathematics%29#Examples

Also, I wanted to point out earlier that:
\sum (a + \sum b) would evaluate to: \sum a + \sum\sum b.

yes, but as matter commented, if
\sum_j a+b
if both a,b involve "j" (the summation index) then it makes sense the sum sign applying to both of them

f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

Stux, I think you misspoke slightly...

"established convention is to give \sum a lower operator precedence than '+'"

yes, I agree.

But then

"\sum a + c should necessarily be understood as (\sum a) + c or c + (\sum a)"

does not make sense.

*Lower* precidence means sum last, and get \sum (a+c).

In the case of

\sum a + \sum b

(without parentheses), the above rule says we have

\sum (a + \sum b)

However, that is *not* always what is meant when the expression is
encountered in the literature! And I don't think one will frequently
see

(\sum a) + (\sum b)

even when that is what was meant. I'd have to say that the convention
that is actually employed in day-to-day math is actually more
complicated than simple operator precidence! But I think on PM, we
should only use one rule to determine where the parentheses: assume
everything in OrderOfOperations, and then add parentheses as needed.

jac,

You know what? You're absolutely right, I definitely did 'misspeak' (If that's a word at all,) about the operator precedence for \sum. I had to think about it for a second! I'm sorry. So yeah it wouldn't be lower precedence. So pretty much the convention employed is a little more complicated than order of operations, but I don't think it's by much:
Generally, multiplications and powers are kept within the \sum, so I'd say (in a preliminary attempt to formalize the convention), it would have a higher precedence than addition but lower than multiplication/division, etc. -- I haven't even touched Integration, etc. The exception would be what drini mentioned: that it would make sense that \sum a_i + b_i is iterpreted as \sum (a_i + b_i); (granted I prefer the latter version cuz its prettier ;) ).
I am curious to know if there is already some kind of paper or documentation formalizing the convention. I wish I could find something like that -- it would be very helpful.

Alright guys, your deep discussions on the scope of summations have borne fruit, although you might need a magnifying glass to see it. In the newly-added entry "proof of the multiplication formula for the gamma function" there is a summation in the exponent in the seventh equation. Thanks to you all, I was VERY, VERY, VERY CAREFUL to enclose the argument of that summation in parentheses lest there be confusion.

Pariuntur montes et nascitur ridicullisimus musculus.

While I'm no mathematician, I remain an interested observer. It seems to me that the example given above for summation of 1/2^k for k=1 to infinity is incorrect.

It is given as 1/1+1/2+1/4+1/8+... but should be 1/2+1/4+1/8+... since the series starts at k=1 and 1/2^1 = 1/2.

If the series had started at k=0, then 1/2^0 would give us 1/1 and thus the series that was printed (1/1+1/2+1/4+...) would have been correct.

Am I correct or did I miss something in analytic geometry?

you're quite right. I fixed it.

dear friends.Whatshould we dowhen the index of a sum is negative?cananybody help?thanks

dear friends.What should we dowhen the index of a sum is negative?cananybody help?thanks

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