Learning Objectives
In this section, you will:
- Expand summation notation.
- Write sums in summation notation.
- Use sigma (summation) notation to calculate sums using summation properties and power sums.
Let’s Get Started…
Archimedes was fascinated with calculating the areas of various shapes—in other words, the amount of space enclosed by the shape. He used a process that has come to be known as the method of exhaustion, which used smaller and smaller shapes, the areas of which could be calculated exactly, to fill an irregular region and thereby obtain closer and closer approximations to the total area. In this process, an area bounded by curves is filled with rectangles, triangles, and shapes with exact area formulas. These areas are then summed to approximate the area of the curved region.
Like Archimedes, mathematicians first approximated the area under the curve using shapes of known area (namely, rectangles). By using smaller and smaller rectangles, we get closer and closer approximations to the area. Taking a limit allows us to calculate the exact area under the curve.
Summation notation makes these calculations easier.
Sigma (Summation) Notation
As mentioned, we will use shapes of known area to approximate the area of an irregular region bounded by curves. This process often requires adding up long strings of numbers. To make it easier to write down these lengthy sums, we look at some new notation here, called sigma notation (also known as summation notation). The Greek capital letter
We could probably skip writing a couple of terms and write
which is better, but still cumbersome. With sigma notation, we write this sum as
which is much more compact.
Typically, sigma notation is presented in the form
where
is interpreted as
Note that the index is used only to keep track of the terms to be added; it does not factor into the calculation of the sum itself. The index is therefore called a dummy variable. We can use any letter we like for the index. Typically, mathematicians use
Let’s try an example.