|
Alternating Series of the Fibonacci
Numbers
A series is called
alternating if it has the form
or
where 
The Alternating Series 
converges if:
(1) ak >
ak+1
(2) 
For the Fibonacci numbers
| 1, |
1, |
2, |
3, |
5, |
8, |
13, |
21, |
34, |
......... |
we have the next
alternating series:
F(n)-F(n+1)+F(n+2)-F(n+3)+F(n+4)-F(n+5)+F(n+6)-F(n+7) +
....
Let us find the alternating 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
operator`s equation
that connects the function
with the series
As will be seen, we usually
deal with the next operator`s identity for the Fibonacci and Lucas
numbers :
and
By using this identity, we
can write :
or
Now, we have the next
alternating series of the Fibonacci numbers:
Alternating series |
Its Sum |
1-1+2-3+5-8+13-21+34-55+... |
1 |
1-2+3-5+8-13+21-34+55-89+...
|
0 |
2-3+5-8+13-21+34-55+89-144+...
|
1 |
3-5+8-13+21-34+55-89+144-233+...
|
1 |
5-8+13-21+34-55+89-144+233-377+... |
3 |
8-13+21-34+55-89+144-233+377-610+... |
5 |
13-21+34-55+89-144+233-377+610-987+...
|
8 |
21-34+55-89+144-233+377-610+987-1047+...
|
13 |
34-55+89-144+233-377+610-987+1047-2034+...
|
21 |
55-89+144-233+377-610+987-1047+2034-3081... |
34
| |
or for Lucas numbers
:
Alternating series |
Its Sum |
1-3+4-7+11-18+29-47+76-123+...
|
-1 |
3-4+7-11+18-29+47-76+123-199+...
|
2 |
4-7+11-18+29-47+76-123+199-322+... |
1 |
7-11+18-29+47-76+123-199+322-521+... |
3 |
11-18+29-47+76-123+199-322+521-843+...
|
4 |
18-29+47-76+123-199+322-521+843-1364+...
|
7 |
29-47+76-123+199-322+521-843+1364-2207+...
|
11 |
47-76+123-199+322-521+843-1364+2207-3571...
|
18 |
76-123+199-322+521-843+1364-2207+3571-5778...
|
29 |
123-199+322-521+843-1364+2207-3571+5778-9349...
|
47
| |
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