# infinity + limit

#### A puzzle

• Gork and Zork are 8 steps away from their spaceship.
• They walk halfway to the spaceship (4 steps).
• Then they walk half the remaining distance (2 steps).
• Then they walk half the remaining distance (1 step).
• Then they walk half the remaining distance (1/2 step).
• Then they walk half the remaining distance (1/4 step).
• If they keep repeating this process.
• Will they ever get to the spaceship?

Did this puzzle surprise you?

How could Gork and Zork ever reach their spaceship?

#### Zeno

The ancient Greek philosopher Zeno maintained that it was impossible for someone to walk to the end of a path because they need to pass through the following distances:

• Half of the path distance,
• half of the remaining path distance,
• half of the remaining path distance,
• and so on

This requires to complete an infinite number of tasks, which, Zeno states, is impossible. This is a paradox, that is, valid reasoning with a logically unacceptable conclusion.

Start with five 16×16 square grid (get the grid paper pdf file).

Shade the grids to represent the fractions 1/2, 1/4, 1/8, 1/16, 1/32

You will end up with something like this:

Imagine doing this forever (you need a good imagination!)

Now imagine cutting out all of the infinite shaded pieces and joining them together to form a single shape. How big would the new shape be?

• Would it fit in your school?
• Would it fit in your classroom?
• Can you hold it in your hands?

#### Geometric series

The sum of fractions we’ve been playing with is a series. A series is the sum of numbers in a pattern. Our pattern is that we get the next term by multiplying the previous term by 1/2.

Some series are arithmetic, like 2 + 4 + 6 + 8, where we add (or subtract) the same number. Some series, like ours, are geometric, where we multiply (or divide) by the same number. Also, some series are finite and some infinite. Our series is infinite.

Can an infinite series have a finite sum? Is there a sum for our series? 