Draw a picture

In a lot of problems, we can make some great headway by drawing a picture.

Example 1

Problem:

Calculate the sum

\[\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots + \frac{1}{2^{2020}}.\]

We think of the sum as gathering up halves of a square

The original area is 1, so our sum will be 1 minus the remaining fraction is \(1 - \frac{1}{2^{2020}}\).