Sum of geometric series


Visual intuition from Art of Problem Solving


  • We have an original element (1)
  • We can make a slightly degraded copy (r). Let’s say r = .95 = 95%. Each time we make a copy, we are left with a smaller version. Then a copy of a copy, and so on.
  • Keep making copies until we run out of juice

The better our reproduction (95% vs 50%) the longer we can go. 1 / (1 - r). Our fidelity is r, our degradation is 1 - r. For example, 95% fidelity = 5% degradation. We can build 20 full copies before the degradation wipes out our chance to make more.
1 / (1 - r) = 1 / (5%) = 20.

Want a stronger intuition for why here. The key intuition is that the better the fidelity, the more copies. The secondary intuition is why it’s exactly 1 / (1 - r).

From the above diagram: It’s really

$$ \Sigma r^n : 1 = 1 : (1 - r) $$

The ratio of 1 to our degradation is the same as the full amount of copies to 1. Maybe it’s easier flipped:

1 to the amount of copies we get = (1 - r) to 1.

Oh! Here’s a thought. If we have 5% degradation, then after generating 20 copies we must have degraded our full amount (i.e., removed our original copy entirely). You have “diluted out” your original amount entirely. A bit like the puzzle with the fly going back and forth. Instead of adding up the series you just say “we will fly for 1 hour, and it goes 60mph, so the total distance 60 miles.”

Primary intuitions:

  1. Taxation
  2. Degredation (fidelity is only r% – how many copies can you make?)

Also see this:

Idea of taxes, then taxes on the tax… and so on. Thinking about the problem “bit by bit” or “all at once”. Great application.