Sum of the Reciprocals of Powers of 2

This explorations asks for alternative ways of finding the sum of the reciprocals of powers of 2. That is, find S when

S is a convergent infinite series. Find multiple ways to show this sum.

Suggestions:

A. This is a geometric series with a common ratio less than 1. Need HINT?

B. Show with a direct proof. Need HINT?

C.

There are 10 kinds of people in the world -- those who know binary and those who do not.

Need HINT?

D. Use similar triangles. Construct a set of squares side by side with each consective square half the side length of the previous. Let AC = 1 and CD will equal S.

Need HINT? (probably not . . .)

E. Other?

Devise activities for helping middle school students develop an

INTUITIVEunderstanding of how this sum gets closer and closer to 2 as more terms are added. That is, how does the fact that this sum is 2 make sense to them?For example, at the right are the first 10 partial sums.

Click HERE for an Excel file

What is the value of knowing alternative proofs of this sum?