Sigma Notation"Sigma notation" is a convenient way of writing large sums. It involves a variable which is present in each term and whose value (an integer) always increases by 1.
 
For example, the sum 1 + 2 + 3 + … + 100 can be written using the variable m and noting that m goes from 1 to 100, by increments of 1.
 
The command to add these terms is written sum_{m=1}^{100} m.
 
The numbers under and over the capital sigma tell us where to start and where to stop.
 
The expression in terms of m in front of the sigma describes the type of terms that are being added.
 
To add the first 10 squares, that is, to denote 1 + 4 + 9 + … + 100, we write sum_{m=1}^{10} m^2.
 
To add the first 10 cubes, that is, to denote 1 + 8 + 27 + … + 1000, we write sum_{m=1}^{10} m^3. I hope you see the difference.
 
In both cases we are told to start at 1 and end at 10. But the first sigma involves m^2 because we are adding squares. The second sigma involves m^3because we are adding cubes.
 
We can get more fancy! Suppose you want to indicate the sum of the reciprocals of the first 100 numbers. Simple! Write sum_{m=1}^{10} frac{ 1}{m}.
 
Now suppose you are m-phobic and wish to use a different letter. You could change sum_{m=1}^{10} frac{ 1}{m} to sum_{k=1}^{10} frac{ 1}{k} without changing the meaning.
 
You could even go Greek and write sum_{alpha =1}^{10} frac{1}{alpha }. The final answer doesn’t involve the letter you use in the sigma expression.
 
 
Sigma notation can be used for sums involving subscripts (the small numbers or letters written lower than the letters they belongs to).
 
In statistics, a collection of 100 scores can be written x_1, x_2, ...., x_{100}, where the subscript tells which score we have.
 
If we want the average of these 100 scores, we can write it as frac{sum_{k=1}^{100} x_k }{100}.
 
The numerator is the sum of the scores. The denominator is the number of scores, namely 100.
 
 
 
 
Here are three important laws involving sigma notation. Since a sigma expression is a sum, the reasons for these laws will be clear (to me!).
 
We represent a sigma sum by sum_{i=1}^n f(i) since the expression to be summed is usually a function of the dummy index, i, like i^2 or i^3.
 
To save typing, we will leave out "i=1" under the sigma and the ‘n’ over it in the first two bullets. I think it took more time to write the last sentence!
sum[f(i) pm g(i)]=sum f(i)pm sum g(i)This looks like we are distributing sigma. In fact, we are merely changing the order in which we add (or subtract) the terms in the sum.
sum [ccdot f(i)]=csum f(i) This says that we can “pull out” a common factor, c.
sum_{i=1}^n C= nCThis says that the constant term C is added n times in the sigma expression.