==> pickover/pickover.18.s <== ------------------------- In article <1992Nov11.221749.129578@watson.ibm.com> you write: : Title: Cliff Puzzle 18: Difficult Nested Roots : From: cliff@watson.ibm.com : Consider the following nested set of square roots. : : ? = sqrt <1 + G sqrt <1+(G+1) sqrt < 1 + ... >>> : : Here, G indicates "Googol" or 10**100. : The "<" and ">" symbols indicate where the beginning and ends of the : the nested roots. : : 1. What is the value for in this infinite set of nested roots. : 2. What is the next term under the root? : Hint: : In 1911, a twenty-three-year-old Indian clerk named Srinivasa Ramanujan : posed the following question (#298) in a new mathematical journal called : the Journal of the Indian Mathematical Society. : : ? = sqrt <1 + 2 sqrt <1+3 sqrt <1 + ... >>> : Doing a n-depth thing-a-ding on this..... n=1 v=1 2 1.732 3 2.236 4 2.5598 5 2.7551 6 2.867 .... 20 2.99999376 .... so I expect that the sum is actually 3. Or in the general case when the 2 (or the G from above) is replaced by m, then the evaluation of the series is m+1. This CAN be shown as follows.... m+1 = sqrt(1+m sqrt(1+(m+1)*sqrt(....)) m^2 + 2m +1 = 1 + m *sqrt(1 + (m+1)*sqrt(...)) m^2 + 2m = m*sqrt(1+(m+1)*sqrt(...)) m+2 = sqrt(1+(m+1)*sqrt(1+(m+2)*sqrt(...)) Thus if m+1 is then sum when the series is based off m, then m+2 is then sum when the series is based off m+1. Since this works for m=2 (as shown above), then it must work for all whole numbers (mathematical induction is such a wonderful thing...) Therefore, the sum with m=G is G+1. The next term, as show above, is (1+(m+2)*sqrt(1+....)) -- Michael Neylon aka Masem the Great and Almighty Thermodynamics GOD! // | Senior, Chemical Engineering, Univ. of Toledo \\ // Only the | Summer Intern, NASA Lewis Research Center \ \X/ AMIGA! | mneylon@jupiter.cse.utoledo.edu / --------+ How do YOU spell 'potato'? How 'bout 'lousy'? +---------- "Me and Spike are big Malcolm 10 supporters." - J.S.,P.L.C.L