Sigma/Summation Notation


I have an assignment due for it on Monday. Could you guys help?

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Sigma notation is a convenient shorthand for writing out functions where you’re adding a bunch of numbers together.

You use a capital sigma, and you write a variable under the sigma and the ranges you’re adding it up from.

Let’s say you want to add every number from 1 to 100 together. You could write it as 1 + 2 + 3 + … + 99 + 100 and assume people know what you mean, or you could write it as

(n = 1) Sigma (100) n

That’s a much shorter way of writing “add n together for the values of n = 1 to n = 100”

Sigma notation comes up in calculus a lot for what are called infinite sums, for example:

(n = 1) Sigma (infinity) 1/2^n

Which is 1 + 1/2 + 1/4 + 1/8 + 1/16 +… and it equals 2

Sigma notation means to take the sum of the result of all integers in a range through a function. Sounds complicated but it works like this:

The large Sigma will usually have a variable and starting number at the bottom and a different number at the top. For example Sigma[x=0:10] says that the Sigma function will be the result of the sum of all results from x=0 to x=10.

Next it will have a function related to the variable. So of there is a function of x f(x) then Sigma[x=0:10] of that function would be f(0)+f(1)+f(2)+f(3)+(…)+f(10).

And that’s really all it is at its core.

f(x) = x^2 + 1

f(1) = 2
f(2) = 5
f(3) = 10

The summation of x^2 + 1 from x = 1 to x = 3 is 2+5+10 = 17

You are just adding up the values for when you evaluate for the starting integer, ending integer, and any integers in between. With sigma notation, x=1 is below the Sigma symbol and 3 in on top.