Summation Notation

Definition

The summation sign, ${\sum }_{i=0}^1$, tells us that we have to add up a collection of expressions and the $i=0$ and the $n$ tell us exactly how to do this.

The $i$ is a dummy variable since it can be changed to whatever letter you want and it won’t affect the actual sum.

For example,

• $(x+y{)}^n={\sum }_{k=0}^n(\frac{n}{k})x^{n-k}{y}^k$ I know.. it looks terrifying. Calm down. Take a breath. I find the best thing to do is try out an example, maybe it’s just me but I can get a grasp of the subject easier this way.

Examples

${\sum }_{i=0}^{3}i$ Again, looks scary. Never fear! The answer is 6.

“What?”

Yes, I heard you say that. Here’s how it works:

${\sum }_{i=0}^{3}i=0+1+2+3=6$ We start at 0, and add our way up to 3.

“Are you serious?”

Yes.

Here’s how I kinda remembered it: the $i$ starts at the bottom number on the sum sign (0) and we just need to get it up to the top number (3). So we go 0.. plus 1.. plus 2.. plus 3. Constantly adding $i$ to itself until we reach that top number.

Now, please note that the little $i$ on the right of the sum sign is telling us HOW to add these numbers: because it’s a singular $i$, we add $i$ to itself. If the sign had an $i^2$ there instead, we would be adding $i^2$ repeatedly until we reach that top number. For example:

${\sum }_{i=1}^3{i}^2=1^2+2^2+3^3=14$

If you’ve ever dabbled in programming of any description, you are most likely familar with variables and that sort of thing. That’s all $i$ is - a simple variable that we are setting, using, and increasing.

Try this one - can you figure it out? = 1^2+2^2+3^3 = 14

${\sum }_{i=1}^{4}i{x}^i$

Answer:

${\sum }_{i=1}^{4}i{x}^i=1^2+2^2+3^3=14$