Loosely related to Luis, Paul and Dr. Dick's topic of mathematical "tricks", here are some interesting real number infinite expansions that can be used to express pi:

(where the value of pi= 3.14159265358979323846...)


pi expression #1:
pi = square root of {6 (1/1 + 1/4 + 1/9 + 1/25 + 1/36 + 1/49 + ...)}

pi expression #2:
pi = 2 {(2/1)(2/3)(4/3)(4/5)(6/5 )(6/7)(8/7)(8/9)...}

pi expression #3:
pi = 4 (1/1 - 1/3 + 1/5 -1/7 + 1/9 - 1/11 + ...)


Note: If you’re trying to verify these three separate expressions by calculator, just realize that in order for the calculation to come out exactly to the value pi, the infinite expansion has to be completed (for every-day life we almost always approximate infinite expansions, but for mathematical statements like these three expressions to work, the real numbers have to be specified exactly).

-Kyle

• Good old summations... where would we be without Calculus? :) - stormcrow - September 4, 2002 - 22:11 UTC
  • Thanks, Mario. Likewise! ( nt ) - cosmosapiens - September 5, 2002 - 03:02 UTC
• puzzling about pi expression 3: obtain QED? - dolphin - September 5, 2002 - 03:14 UTC
  • Sorry, I didn't understand a word of that - stormcrow - September 5, 2002 - 19:22 UTC
    • pi - dolphin - September 6, 2002 - 07:37 UTC
  • Great imagination - luis_hamburgh - September 5, 2002 - 23:44 UTC
    • A little goes a long way Luis - dolphin - September 6, 2002 - 05:53 UTC