HOMEWORK #3 : Geometric Series and Xenos Paradox
(due Fri-3/3)

An infinite sum of i terms that follows the pattern a * r ^ i might look like this:

• S = a + ar + ar²+ ar³ + …
  

This sum not only converges upon a finite number, but actually attains that number. Here is an ugly, imprecise proof: Suppose you multiply both sides of the above bulleted sum by r:

• Sr = ar + ar²+ ar³ + …
  

Subtract the second bullet from the first:

• S - Sr = (a + ar + ar²+ ar³ +…) – (ar + ar²+ ar³ + …)
  

All the terms on the right except a cancel and you are left with:

• S – Sr = a
• S (1 – r ) = a
• S = a / (1 - r)

This is related to Xeno’s Paradox when 0 is less than r is less than 1, because for every power incremented larger, r becomes a smaller number and the step you take toward the destination is smaller and smaller, but as the formula above assures, the destination will be attained.

For the three specific sums we were asked to calculate, the above is still true.

The sum (o less i less inf) of 1/(2^i) = 2.

The sum (1 less i less inf) of 1/(2^i) = 1.

The sum (2 less i less inf) of 1/(2^i) = 1/2.

Analytically, each progressive start value for i greater than the last contains one term fewer and is therefore a smaller sum by a proportional value. If the base were 3, it would still get smaller, but it would not half each time. The results would be... 3/2, 1/2, 1/6, which is a third smaller each time.

***********I consider this homework set to be complete***********