# What is the summation operator used for in Mathematics?

I am quite experienced in programming and I know that summation in math is similar to a for loop: It runs the specified operations for $x$ amount of times. What I want to know is how this is useful in mathematics. I don't see what effect recursively running operations in math could yield that could be any different from doing so manually. Why does the variable that is incremented by one each loop have to do so? I'm currently in 11th grade and my class has not yet touched upon this topic, but I have seen it across this site a plethora of times and would like to know more about it.

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Certainly someone will come up with a more satisfying answer, but if nothing else, it's a matter of compactification: instead of writing, say, $a_1 + a_2 + a_3 + \dots + a_n$, we write $\sum_{i = 1}^n a_i$, which is shorter. It may appear irrelevant here, but within big equations, it gets cumbersome to keep writing the dots and the summation becomes important. – student Feb 11 '12 at 1:54
Btw, "for" loops don't have to increment by $1$, and neither do (the index in) sums. – The Chaz 2.0 Feb 11 '12 at 2:09
As a background note, there are things called "operators" in mathematics, and the summation symbol is not an operator. (If one wanted to they could view it as a functional...) – anon Feb 11 '12 at 2:09
@TheChaz I was referring to the Sigma function where a given variable $n$ is incremented by one each time. – David Feb 11 '12 at 2:26
I know! I was referring to, say, the index of sigma in the divisor function. – The Chaz 2.0 Feb 11 '12 at 3:01

There are many formulas in which an operation is repeated over and over. Addition and multiplication are so common that we have corresponding summation and product notations. These are convenient ways to give compact expressions for various formulae.

For example: Instead of writing $1^2+2^2+3^2+\cdots+n^2$ we can write $\sum\limits_{i=1}^n i^2$

The first place most students see summation notation used in any serious manner is in calculus when Riemann sums are defined. A Riemann sum approximates areas under curves using rectangles.

For example: If we want to approximate the area under $y=x^2$ where $0 \leq x \leq 3$ using 6 rectangles. Then we break the interval $[0,3]$ into 6 pieces: $[0,1/2]$, $[1/2,1]$, $\dots$, $[5/2,3]$. If we use $y=0^2$ as the height of the first rectangle, $y=(1/2)^2$ as the height of the second, and so forth, then noting that the width of each of these rectangles is $1/2$ we have that the area under the parabola is approximately $\frac{1}{2}(0)^2+\frac{1}{2}(1/2)^2+\frac{1}{2}(1)^2+\frac{1}{2}(3/2)^2+\frac{1}{2}(2)^2+\frac{1}{2}(5/2)^2$. Summation notation lets us write this much more compactly as $\sum\limits_{i=0}^5 \frac{1}{2}\cdot \left(\frac{i}{2}\right)^2$. Now what if we wished to approximate with $n$ rectangles? We have $\sum\limits_{i=0}^{n-1} \frac{3}{n}\cdot \left(\frac{3i}{n}\right)^2$ (nice and compact).

Similarly mathematicians use product notation to express repeated multiplications.

For example: $\prod\limits_{i=1}^4 i^2=1^2\cdot2^2\cdot3^2\cdot4^2$.

Or you may be familiar with the factorial function: $n! = \prod\limits_{i=1}^n i = 1\cdot 2\cdot 3 \cdots n$

By the way, the analogy with for loops is a good one. However, for loops allow you to repeat a plethora of operations. Summation and product notations are far more specialized (summations and products can both be computed using for loops). Also, not all for loops increment the index by 1 each time. Most programming languages allow for more complicated increments. In the same way summations and products are sometimes done over sets (other then ${1,2,\dots,n}$).

For example: If $S = \{1,2,4,8\}$, then $\sum\limits_{x \in S} \frac{1}{x} = \frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8}$.

In calculus (often second semester calculus) one learns about series where you are essentially summing over infinite sets [summing over infinite sets gets a little tricky -- there are issues of convergence and problems when shuffling the order of the summation around].

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N.B. summation over sets of indices corresponds to the foreach() construction seen in some programming languages. – J. M. Feb 11 '12 at 2:37
@Bill : On this website you should avoid saying things such as "In calculus II one learns that..." because programs differ from one place to another and the beginning of your sentence becomes irrelevant. – Patrick Da Silva Feb 11 '12 at 14:36
I understand. However, the OP referred to the "11th grade" which means he is almost certainly in the US. – Bill Cook Feb 11 '12 at 15:06

What if you want a summation where the number if terms is variable? For example, $$f(n)=1+2+3+\cdots+n=\sum_{k=1}^nk .$$

When you get into infinite series you can't just do an infinite number of arithmetic operations by hand, so mathematics (analysis in particular) devised what's called a limit with which we can talk about infinite sums, and in order to do this talking concisely we need notation that helps us to refer to partial sums in a speedy and easy-to-understand manner. Series expansions are ubiquitous in any area of mathematics based in analysis (ranging from differential equations to geometry to number theory to combinatorics to you-name-it).

As another practical utility, sometimes the summands (the individual quantities being added) come in a pattern that is not easy to spot in the wild and we can forego the obstacle of our readers trying to figure out what these patterns are by simply writing in summation notation, where each of the summands can be written as an explicit function of the index variable and no guessing is needed.

Finally, summations of summations are cumbersome to write and extremely difficult to parse in notation-naked writing, but have straightforward notation and interpretation with the $\sum$ symbol.

Try writing $$\sum_{a=1}^{100}\sum_{b=1}^a \sum_{c=1}^b \sum_{d=1}^c \big(a-b+c-d\big)$$

without summation notation!

Another utility I didn't think of while originally writing is that sometimes we have summations that are indexed by some set $X$ of numbers or other objects and it is unnatural to view this set as a sequence. For example, for a integer $n$ we could define the sum

$$\sum_{\gcd(n,m)=1}\frac{1}{m^2}.$$

This is an infinite series of the reciprocals of the squares of all numbers coprime to $n$. The set of naturals coprime with $n$ can be viewed as a sequence but trying to find a way to enumerate them as $f(1),f(2),\dots$ is superfluous and only serves as distraction from the underlying concepts. (Though convergence is another issue..)

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"Why does the variable that is incremented by one each loop have to do so?"

It does not, but that is simply our convention. If we would like the sum to increase by two each time we would not change the meaning of

$$\displaystyle\sum_{i=1}^k a_k$$

but rather we simply define $b_k = a_{2k}$ and compute the sum

$$\displaystyle\sum_{i=1}^k b_{k}.$$

We do this because it's much more general to keep the increment increasing by one and then letting the index of the sequence we are summing over correspond to some sequence itself!

The summation has a lot of applications, but I would like to point out that not all summation operators can even be expressed in the form $\displaystyle\sum_{i=1}^k$. It turns out there are sets that are so infinite that they cannot be expressed in the form $\{a_1,a_2,\ldots\}$ where you use $1,2,\ldots$ to count them. Such a set is called uncountable (for example, the real numbers are uncountable).

This suggests that we need an alternative definition of the summation for a sum over an uncountable set. Let $I$ be some set and for every $\lambda$ in the set $I$, let $\{a_{\lambda}\}$ be some object. We use the notation

$$\displaystyle\sum_{\lambda \in I} a_{\lambda}$$

to denote the sum over all the elements $a_{\lambda}$ "indexed by $I$". This notation is general enough to cover both the sums you know about (the "increment by one type"): for example let $I = \{1,2,3,4,5,6\}$, then

$$\displaystyle\sum_{\lambda \in I} a_{\lambda} = \displaystyle\sum_{\lambda=1}^{6} a_{\lambda}.$$

So, although you can't formalize "what's the sum of all positive real numbers?" in the form $\displaystyle\sum_{i=1}^{\infty} a_i$ (since the positive real numbers are uncountable), you CAN represent it as $\displaystyle\sum_{\lambda \in \mathbb{R}^+} \lambda$, where $\mathbb{R}^+$ represents the set of positive real numbers.

What's the use? Well, in terms of sums of real numbers it's not much use, as the uncountable sum of positive real numbers is always infinity. Its actual use is in advanced mathematics where we think about such sums of various kinds of objects, where "sum" doesn't mean "add up numbers" but "an operation that behaves enough like addition that we call it addition". It takes much time to explain, but if you really want the name of an area of math that uses uncountable sums, the one I know of is "nonseparable Hilbert spaces", but I'm sure there's uncountably many more!

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It is just a notation to easily represent what is being implied by the expression. Imagine you want to say $x =$ sum of the first 50 natural numbers, then you can express that by saying $1 + 2 + 3 .... + 50$ or use the summation operator in which $n$ goes from $1$ to $50$ for convenience. Similarly, there is product operator too which looks like pi but has suffixes similar to a summation operator although there may be a better name for it.

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