Sigma notation provides a way to compactly and precisely express any sum, that is, a sequence of things that are all to be added together. Although it can appear scary if you’ve never seen it before, it’s actually not very difficult. Here’s what a typical expression using sigma notation looks like:
We would read this as “the sum, as k goes from a to b, of f(k).” In plain English, what this means is that we take every integer value between a and b (inclusive) and substitute each one for k into f(k). This results in a bunch of values which we add up.
Let’s go through each part of that and see what they mean in more detail:
- : this is a capital sigma, the eighteenth letter of the Greek alphabet. It is not an ‘E’! Sigma corresponds to the English letter ‘S’; ‘S’ is for ‘sum’.
- k: The k on the left side of the equals is called the index variable or the index of summation, or sometimes just the index. It will take on all the integer values between a and b (inclusive).
- a, b: a is the starting index and b is the ending index.
- f(k): this is the expression that describes each term in the sum. For each value of k between a and b, f(k) will be some value which gives one term in the sum.
If you’re still confused, don’t worry; an example should make things clear!
See how that works? We took every value of k between 2 and 5 inclusive, and substituted each into the expression ; then we added everything up.
As a bonus, once you understand sigma notation, you understand Big Pi notation for free: a Big Pi () works exactly the same as a Big Sigma, except it denotes multiplication instead of addition (‘P’ is for ‘product’). For example: