==> 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 /
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