OFFSET
1,1
COMMENTS
How Johann Jakob Ballmer found his formula in 1885 by analyzing and manipulating the ratios of these data:
r(1) = a(1)/a(1) = 1,
a(2)/a(1) = 1.349961..., rounded: r(2) = 135/100 = 27/20,
a(3)/a(1) = 1.511996..., rounded: r(3) = 1512/1000 = 189/125,
a(4)/a(1) = 1.599996..., rounded: r(4) = 16/10 = 8/5,
a(5)/a(1) = 1.6530647..., r(5) = 81/49 = 2-1/(3-1/(9-1/2)), derived from a(5)/a(1) = 2-1/(3-1/(9-3095/6216)) when replacing 3095/6216 by 1/2;
the multiplication of these fractions by 5/36 is the key trick to get more handy figures to see eventually increasing squares in the denominators by an appropriate expansion:
b(1) = r(1)*5/36 = 5 / 36,
b(2) = r(2)*5/36 = 3 / 16,
b(3) = r(3)*5/36 = 21 / 100,
b(4) = r(4)*5/36 = 2 / 9,
b(5) = r(5)*5/36 = 45 / 196;
... b(1) .|.... b(2) ..|.... b(3) ..|.... b(4) ..|.... b(5),
... 5/36 .|.... 3/16 ..|... 21/100 .|.... 2/9 ...|... 45/196,
... 5/36 .|... 12/64 ..|... 21/100 .|... 32/144 .|... 45/196,
(9-4)/9*4 |(16-4)/16*4 |(25-4)/25*4 |(36-4)/36*4 |(49-4)/49*4,
this last step was the crowning achievement: the discovery of the pattern (x-y)/x*y,
b(n) = ((n+2)^2 - 4)/(4*(n+2)^2) = 1/4 - 1/(n+2)^2;
REFERENCES
R. Taschner, Der Zahlen gigantischer Schatten, Vieweg 2005, 137-143.
LINKS
Science Trek, Balmer Formula
Eric Weisstein's World of Physics, Balmer Formula
Wikipedia, Empirical formula
Wikipedia, Johann Jakob Balmer
Wikipedia, Robert Wilhelm Bunsen
Wikipedia, Gustav Robert Kirchhoff
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Reinhard Zumkeller, Dec 22 2006
STATUS
approved