34
THE GENERALIZED FIBONACCI OPERATOR
Nov. 1968
b e c o m e s a r b i t r a r i l y l a r g e a s x a p p r o a c h e s infinity and t h e r e f o r e cannot have
r - 1
a s a limit.
T h i s c o n t r a d i c t s o u r a s s u m p t i o n that r > 1.
That
Him r
= 2
x-^0 x
follows i m m e d i a t e l y from the fact t h a t
P0(A) = X2 - 2A .
REFERENCES
18
L. W. Cohen and Nelson Dunford* " T r a n s f o r m a t i o n s on Sequence S p a c e s /
Duke Math, J e , 3 (1937), 689-701.
2.
A. E„ T a y l o r , Functional A n a l y s i s , J e Wiley.
• • * •
•
A CURIOUS P R O P E R T Y OF A SECOND FRACTION
M a r j o r i e Bicknell
A. C. Wilcox High Schools Santa C l a r a ,
California
In the A p r i l , 1968 Fibonacci Q u a r t e r l y (p. 156), J . W l o d a r s k i d i s c u s s e d
some p r o p e r t i e s of the fraction 878/323 which a p p r o x i m a t e s e. C o n s i d e r the
a p p r o x i m a t i o n of rr c o r r e c t to six d e c i m a l p l a c e s given by 355/113 =
3.141592 + . The sum of the digits of the n u m e r a t o r i s 13, and of the d e n o m i n a tor, 5. 1 3 / 5 = 1 + 8/5, o r one added to the b e s t approximation to the "Golden
R a t i o n using two one-digit n u m b e r s . Also,
355 = 300 + 55
113
100 + 13 J
w h e r e 55 and 13 a r e Fibonacci n u m b e r s .
Taking 355/226 a s an a p p r o x i m a t i o n of TT/2 l e a d s to the o b s e r v a t i o n that
355 = 377 - 22
226
233 - 7
w h e r e 377/233 a p p r o x i m a t e s the golden r a t i o and 2 2 / 7 a p p r o x i m a t e s n, and
377 and 233 a r e Fibonacci n u m b e r s .