|
Divergent Series of the Fibonacci
Numbers
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"In his first paper on the Calculus
(1669), Newton proudly introduced the use of infinite series
to expedite the processes of the calculus... As Newton,
Leibnitz, the several Bernoullis, Euler, d'Alembert, Lagrange,
and other 18th-century men stuggled with the strange problem
of infinite series and employed them in analysis, they
perpetuated all sorts of blunders, made false proofs, and drew
incorrect conclusions; they even gave arguments that now with
hindsight we are obliged to call ludicrous."
( From:
"Mathematics: The Loss of Certainty" by Morris Kline
) |
A series is a sum of terms
specified by some rule. A series usually has an infinite numbers of
terms.
Fibonacci ( 1170 - 1240 ) discovered a sequence of
integers in which each number is equal to the sum of the preceding
two introducing it in terms of modeling a breeding population of
rabbits. This sequance has many curious and remarkable properties
and continues to find application in many areas of modern
mathematics and science.
When we have a list of Fibonacci
numbers,
| 1, |
1, |
2, |
3, |
5, |
8, |
13, |
21, |
34, |
......... |
the first
operation that we might think of is to sum them. It is
intuitively trivial to define the sum of finitely numerous Fibonacci
numbers, but how should we define the sum of infinitely numerous
Fibonacci numbers :
A
sequence {F(n)} is called increasing (
resp.decreasing) if we have
F(n)<F(n+1); (resp.
F(n)>F(n+1))
Fibonacci sequence is
increasing.
A sequence {F(n)} is called convergent if
a certain number F, called the limit of
{F(n)}, exists.
A
sequence is called divergent if it is not convergent. As
Fibonacci
sequence is divergent.
We assume that all divergent series are
convergent.
In fact Fibonacci series converges, but the
exact value of the sum proves hard to find. The continual processes in Nature are qualitatively transformed
in natural discrete points of Zero and Infinity. The transformation of Quantity into Quality is defined by the transformation
of the infinite value sums into the finite - qualitative ones. Probably, all the sums of divergent and alternating series have properties of mathematical operators.
Let us find the sum of all Fibonacci numbers! We shall now
introduce operator of the finite differences that connects the
function
with the
function
It is easy to verify
that
where D is operator of
differentation.
We shall now introduce
inverse operator of the finite differences
that connects the function
with the series
As will be seen, we usually
deal with the next operator`s equations for the Fibonacci and Lucas
numbers :
and
By using this equations, we
can write :
or
Now, we have the next series
of the Fibonacci numbers:
Divergent series |
Its Sum |
1+1+2+3+5+8+13+21+34+55+... |
-1 |
1+2+3+5+8+13+21+34+55+89+...
|
-2 |
2+3+5+8+13+21+34+55+89+144+...
|
-3 |
3+5+8+13+21+34+55+89+144+233+...
|
-5 |
5+8+13+21+34+55+89+144+233+377+... |
-8 |
8+13+21+34+55+89+144+233+377+610+... |
-13 |
13+21+34+55+89+144+233+377+610+987+...
|
-21 |
21+34+55+89+144+233+377+610+987+1047+...
|
-34 |
34+55+89+144+233+377+610+987+1047+2034+...
|
-55 |
55+89+144+233+377+610+987+1047+2034+3081... |
-89
| |
or for Lucas numbers
:
Divergent series |
Its Sum |
1+3+4+7+11+18+29+47+76+123+...
|
-3 |
3+4+7+11+18+29+47+76+123+199+...
|
-4 |
4+7+11+18+29+47+76+123+199+322+... |
-7 |
7+11+18+29+47+76+123+199+322+521+... |
-11 |
11+18+29+47+76+123+199+322+521+843+...
|
-18 |
18+29+47+76+123+199+322+521+843+1364+...
|
-29 |
29+47+76+123+199+322+521+843+1364+2207+...
|
-47 |
47+76+123+199+322+521+843+1364+2207+3571...
|
-76 |
76+123+199+322+521+843+1364+2207+3571+5778...
|
-123 |
123+199+322+521+843+1364+2207+3571+5778+9349...
|
-199
| |
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