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SUMMATION OF INFINITE FIBONACCI SERIES
BROTHER ALFRED BROUSSEAU
St. Mary's College, California
In a previous paper s a well-known technique for summing finite or infinite series was employed to arrive at a number of summations of Fibonacci and
Lucas infinite series in closed form [1]. This work is rewarding but in reality covers only a limited portion of the possible infinite series that can be constructed.
Starting in general with an arbitrary Fibonacci or Lucas infinite
s e r i e s , the probability that it has a closed sum is relatively small.
One need
only think of the sum of the reciprocals of the Fibonacci numbers themselves
which to date has not been determined in a precise manner.
In the face of this situations what remains to be done? The present a r t i cle attacks this problem by attempting to accomplish two things.9 (1) Determining the relations among cognate formulas so that formulas can be grouped into
families in which all the members of one family are expressible in terms of
one member of the family and other known quantities; (2) Replacing slowly converging sums by those that converge more rapidly.
The combination of these two efforts has this effect
A29 A35
a 35 «* e ,
e ee
Given families A lf
» whose members a r e expressible in terms of summations a * ^ ?
respectively.
Then if these quantities a. can be related to other
quantities a! which converge more rapidly, the problem of finding the summations in the various families is reduced once and for all to making precise
determinations of a very few summations a! which can be found in a reasonably small number of steps*
Such is the program* The purpose of the article is to give an illustrative
rather than an exhaustive treatment.
The investigation? moreover, will be
limited to infinite series of the type:
00
Y^
1
/ j F F
F
F
*e * F
~¥
n n+kj n+k2 n+k3
n+k r
n~~ i.
&•
•
o r of the form 9
143
144
SUMMATION OF INFINITE FIBONACCI SERIES
[Apr.
vll-l
F
n=i
F
n
F
••• F
n+kj n+k2
n+k r
with all the Fibonacci numbers in the denominator different and the k. positive*
NOTATION AND LANGUAGE
To compress notation, the expression (F )
F
n
F
n-11
F
0
n-2
•• e F
will mean
,- .
n-r+1
If there are k Fibonacci numbers in a denominator, we shall speak of this as
a "summation of the k
degree. TT
CONVERGENCE OF THE SUM OF FIBONACCI RECIPROCALS
We shall begin by establishing the fact that the sum of the reciprocals of
the Fibonacci numbers:
00
n=l
converges. This in turn will be sufficient in itself to enable us to conclude to
the convergence of all sums of our two types since their terms are less than or
equal to those of this series.
Using the roots of the equation x2 - x - 1 = 0, namely,
r
'1 + *s/5
= __.
,
and
s
we have
(l)
n
n
Fn = £ - ^ S n
N/5
1 - *s/5
= —.———
,
1969]
SUMMATION OF INFINITE FIBONACCI SERIES
Now s = - r
. Hence when n is odd,
l/F
145
Vs/r11
<
and when n is even, it can be shown that
<V5/rn"1
l/F
This follows since the relation for n even,
rn~
> r~ n ,
for n = l ,
or finally
r - 1 > r~
n
.
rn - r~ n
rn~
leads to rn -
which is certainly true for n _>_ 2?
r - 1 = r" .
Thus in either case
l/F
<^ / s / r 1 ,
for
n >2
Hence
CM
00
£vF n <£V^- T
(2)
n=l
<s/5
•T7F
n=l
Since the summation of positive terms has an upper bound, it follows that it
must converge.
RELATIONS AMONG SECOND-DEGREE SERIES
Essentially, there is only one first degree series of each type in the
sense defined in this treatment, so that the first opportunity to relate series
comes with the second degree. Here we have a special situation inasmuch as
the alternating series can all be evaluated, the final result being:
00
(3)
n-1
(-I)"
F F
n=l
nn+k
r being defined as before.
F,
kr"
1
-EVA
The proof is as follows.
146
[Apr
SUMS OF INFINITE FIBONACCI SERIES
00
i2-^L
F
n-l/Fn ~
F
n+k-l/Fn+kJ
n=l
K
n
j=l
m=n-k+
lira
n+k-l/
m+k
EwF3-kr_1
j=i
But the initially given summation also equals
00
F
n-1
F
n=l
- F F
n+k
n n+k-1
F F
n n+k
^-(-i)V
n=l
F
F
n
n+k
Equating the two values and solving gives relation (3).
The non-alternating series of the second degree has closed formulas for
the summation
F
n
F
n+k
n=l
when k is even. For the case k = 2,
00
F
n=l
n F n+1
F
n+1 F n+2
1969]
SUMS OF INFINITE FIBONACCI SERIES
147
But this likewise equals
2n =^l F n Fn+2
s o that
00
(4)
>
i r
n=l
n
^ — = 1
n+2
F o r k = 4 5 the d e r i v a t i o n i s a s follows.
no
Z-/
n=l
oo
F
n
F
n+2
F F
L-d
n n+4
n=l
E
- ^ - I F , , - S6 F
F , .
*n+4 n+2
"
"
x
x
n+4
n = 1 "n n+2
=
^
F
V^
^—^
F
n
F
n
n+2
F
00
" 2 ^- FF n-+T2 IF n + 4
n=l
00
1 ' 2
1 *3
/ >
• ••^n-=Tl
F
F
n + 2
i
x.
n
Solving for the d e s i r e d s u m m a t i o n ,
00
5 y_i_
F
F
^V n n+4
n=l
00 ,
=
2
V-J_
F
F
Z-' n n+2
n=l
5/6
n+4
148
SUMS OF INFINITE FIBONACCI SERIES
[Apr.
so that
ex?
(5)
/
*-4
n=l
— ?
F
n Fn+4
= 2/3 - 5/18 = 7/18 .
The process can be contained yielding:
(6)
Y ^
^mm 4
n=l
F
/
= 143/960
n n+6
and an endless series of formulas with a closed value.
For k odd
00
GO
00
2 ¥
Zjv^TXf
7^
n=l
n==l
=
F F
Xf
n + 1 n+3
n=l
00
•A+Zw2
= 1/2+1 1/2
-
-
Therefore
00
oo
E
2^^^= ^ " - - 1 / 4
l
(7)
=
I P
1_
F F ^
2 / J F F
—7
*—7 n n+1
- n n+3
n
n=l
n=l
It is possible to proceed step-by-step to other formulas in the series.
A
Thus
^ 00
21
00
°°
/ J F n F n+3
^ ~ 52-jF
iS
±K
*—T
—rf n Fn+5
n=l
n=l
7 ^ F n+3 'F n+5
,,^—7
n=l
CO
=
._2L__i
1-2
1-3
L.+V
2-5
I
^ ^ - F n Fn+2
n=l
1969]
SUMMATION OF INFINITE FIBONACCI SERIES
149
Hence
00
^/
00
J
F F ,.
1/10 + 1/5(1/2 + 1/3 + 1/10 - 1)
5
F _,^/ J F
n n+i
n n+5
or
(8)
X
00
=
i7/i5
2 ^b; E "^ - ° •
n
^ 1
n+5
n n+1
^
In summary, for second-degree summations of the given types, apart from the
results in closed form, the summations
00
i;
n=l
F F
n n+k
with k odd are all expressible in the form
00
• + b £»^r
n=l
where a and b are rational numbers,
AUXILIARY TABLE
In the work with these summations, the formula that i s being employed
to arrive at Fibonacci numbers which are to be eliminated is:
<9>
F
kW r
" Fk+r Fk+n =
^ " V n
or
<10>
F
n = ^ r ~ [Fk F k + n + r " F k + r
F
k+n]
150
SUMMATION OF INFINITE FIBONACCI SERIES
[Apr.
Rather than use this for each instance it is found to be more convenient to make
a table which indicates factors a and b in the relation:
(11)
F
n
The quantities F
= a F ,
+ b F r
n+k
n+j
, are at the right; the quantities F
. a r e at the top. The
tabular values for any given pair are a,b in sequence. Thus, to express F
n
in terms of F + „ and F 2 , the quantities a and b are - 1 / 3 and 8/3,
respectively, so that
F n = (-F n+6
^ + 8F n+2"
AO )/3
Similarly, to express F + o in terms of F
+g
and F
subscripts is relative, we take the table values for
F
5,
+5
since the shift in
and F + ~.
Hence
F n+3
^ = (-F n+8
^ + 5 F n+5 ^i/l
'
Table I
QUANTITIES a,b IN FORMULA (11)
F
F
n+k
F
F
n+2
F
F
J ..
n+3
n+3
n+4
F
n+5
F n+6
±*
F
*n+7
n+1
1,-1
F
F
n+2
n+3
F
n+4
F
n+5
1.-2
-1,2
1/2(1, -3)
-1,3
2,-3
1 / 3 ( 1 , -5)
1 / 2 ^ 1 , ,5)
2,-5
-395
1 / 5 ( 1 , -8)
l / 3 ( - l 1 ,8)
1/2(2,-8)
-3,8
5,-8
1/8(1, -13)
1/5(01,13)
1/3(2,-13)
l/2(-3,13)
5,-13
THIRD-DEGREE SUMMATIONS
For third-degree summations
F
n+6
-8,13
1969]
SUMMATION OF INFINITE FIBONACCI SERIES
there is one which has its sum in closed form.
151
The derivation follows.
00
n=l
F
F
n
F
n+1 n+2
F
F L+2Fn+3 J
n+1 n+2
2
But this also equals
00
n=l
F
n+3
- F
n
n+1
= 2
°W
Hence
PC
(12)
n=l
I
F F
F
n n+2 n+3
4
To find
Ty
"P
n
n=l
"P
n+2
n+4
in terms of
00
n=l
^wT
we use this result, arranging coefficients so that we obtain F
ator and then eliminate it from the denominator.
Thus
in the numer-
152
SUMMATION OF INFINITE FIBONACCI SERIES
2F
n+4-3Fn+3
Y
+S -
n=l
F
F
F
F
_y
FA
_n_
F
F
*—*
n nH-2
n
—f "n
"n+2 "n+3
A
nn "n+2
n+?. "n+3
n + 2 nn+4
+4
[Apr.
F
n+4
n=l
_ 00
OP
n=l
*
6
n=l
Solving for the desired summation,
00
^\ ~"*
—7n=l
_ « _
11
i
n n+2xn+4
7
7
xu
1
l \^^
^ ^
°^—'
n=l
vx
11
n+2' 3
6
The procedure is similar at each step. There are two formulas (a) and (b) to
appropriat coefficients; a certain F
be combined with appropriate
mula (c) is obtained, either
n=l
is eliminated; a for
6
or one which has previously been expressed in terms of this quantity.
It would occupy altogether too much space to present even a small portion of the derivations. The sequence of steps, however, can be indicated by
giving the denominators in the summations (a), (b), and (c) and between (b) and
(c), the quantity F
which was eliminated.
summation is the same as (b) in this table.
The denominator of the desired
1969]
SUMMATION OF INFINITE FIBONACCI SERIES
153
Table II
SCHEMATIC SEQUENCE FOR THIRD-DEGREE SUMMATIONS
oo
1
T
if i f
F
n=l
D e n o m i n a t o r (a)
F
F
n
F
F
F
F
n+2
F
n
F
n
n
F
n
F
F
F
n+3
n+1
n+1
n+1
F
n+2
n
n+3
F
n+3
n+4
n+3
F
F
F
F
n+3
F
F
n
F
F
F
n+4
F
F
F
F
n+4
F
F
n+4
n+5
F
n
F
n+2
n
F
n+a
n+b
F
D e n o m i n a t o r (b)
n+4
F
n
n
F
F
F
n
F
n+2
F
n
n
n
F
n
n
n+3
n+3
F
F
n+1
n+1
n+1
n
n
n+2
F
F
F
F
F
n+3
F
n+4
F
F
n+5
F
n+4
F
F
n+5
F
n+2
F
F
n+4
F
F
F
n+5
F
n+4
D e n o m i n a t o r (c)
r
F
n
F
n+5
F
n
F
n
n
n
n
where
00
1
n+s
F
=
n+t
n+3
F
n+4
n+3 F n + 4
F
n+5
C +
n+2
n+1
n+3
F
n+4
F
n+1 F n + 4
F
n+5
F
n+5
F
n+6
n+3 F n + 4
F
n+5
F
n+6
n+2
F
J
n+3
n
F
F
n+1 F n + 3
F
n
00
/.J?
^—'
n=l
n+5
F
F
and so on.
The results can be summarized in the form
(14)
F
n+2
F
n+6
n+4
F
n
F
n+6
F
n+2
F
n+4
SUMMATION OF INFINITE FIBONACCI SERIES
154
[Apr.
Table HI
1ST ANT'S c AND d FORGIVEN s AND
i
t IN FORMULA (14)
s,t
1,3
c
1
-1/4
-7/36
-71/450
2/3
7/15
1,6
-509/4800
3/10
1,7
2,3
-11417/162240
5/26
0
2,4
1/4
7/18
-1/3
2,5
71/300
-1/5
2,6
509/2880
-1/6
2,7
11417/101400
-7/65
3,4
1/3
3,5
-5/36
-67/300
3,6
-269/19 20
4,5
19/225
1/4
-1/15
4,6
407/2880
-1/6
1,4
1,5
2/5
It should be apparent without formal proof that any summation of the third degree with positive t e r m s can be expressed in the form given by (14).
A prac-
tical conclusion follows: It is only necessary to find the value of one summation
n=l
6
which can be done once and for all to any desired number of decimal places.
Thereafter for formulas related to this summation their values can be found
with a minimum of effort to any desired number of places within the limits
established for the one basic formula.
This method of relating a number of formulas to one formula can be continued to higher degrees though the complexities become greater.
ple 9 for seventh-degree expressions:
For exam-
1969]
SUMMATION O H INFINITE FIBONACCI SERIES
xn-l
£
n=l
155
<-ir
n F jj. - n+k.
i=l
i
we can p r o c e e d s t e p - b y - s t e p a c c o r d i n g to the following table.
(a), (b) , (c) and F
The quantities
have the ssame
a m e m e a n i n g a s in T a b l e II. T h e d e s i r e d s u m -
m a t i o n i s indicated by an a s t e r i s k .
T a b l e IV
SCHEMATIC SEQUENCE FOR SEVENTH-DEGREE SUMMATIONS
WITH ALTERNATING TERMS
D e n o m i n a t o r (a)
(F
D e n o m i n a t o r (c)
D e n o m i n a t o r (b
<W 7
*<F _ ) F J ._
n+5 6 n+7
*(F
) F
v
n + 5 ; 6 n+8
*(F
) F
1
n + 5 ; 6 n+9
n+6 \
^n+6^
(F n +^6 )'
7
(F ^ ) F MQ
n+6 6 n+8
(F n+6
^ ) g F n+9^
and so on.
(F n+6
^ )
(F
^n+S^n+T
7
^n+5^
F
n+7
(F
"n+5
*(Fn+4)5Fn+7
(F n+6
^ )
(F
(F
7
.c)
n+6
and so 7on.
n+7
"n+5
^11+5^
n+5
)
6
F
F
n+8
n+9
n+5
^R) F A_
n+5 g n+7
^n+S^n+S
"n+4
7
"n+4
GENERAL CONCLUSION
It i s p o s s i b l e to e x p r e s s all s u m m a t i o n s
00
n=l
n Fn+k.
i=l
F
F
"n+5
n+b\ n+7
and s o on.
F
n+4)5Fn+6
n+8
(F
(F
(F
n+5)6Fn+9
^n+5^
n+6\
^n+6^
and s o on.
"n+5
* n+4* 5 - n+6 F n + 8
*(F
) F
F
1
n + 4 ; 5 n+6 n+9
F
(F
n+4
F
)
5
F
n+7
F
n+8
F
n+9
?Fn+3>4 < W ,
(F
n + 3 * 4 F n + 5 Fn+6
F
n+8
156
SUMMATION OF INFINITE FIBONACCI SERIES
[Apr.
in the form
00
a +b
2 ^ (Fn + r x )
n=l
r
where a and b are rational numbers; and all summations
y* (-if-1
r
n=l .n . F.n+k.
i=l
l
in the form
00
c +d
n=1
i-if-1
(F
n+r-l)r
where again c and d are rational numbers.
The limitation of this approach is that it is not possible to proceed directly in one step to this final result as a rule. It is necessary to go through
a series of formulas and should the desired summation be remote from the
final objective, this could be a long operation.
Once, however, the various
formulas have been linked to the one formula, the problem of calculating these
summations becomes relatively simple.
This concludes the discussion of linking formulas of the same degree.
We now proceed to a consideration of expressing a summation of lower degree
in terms of one of higher degree so as to secure more rapid convergence. But
first, formulas will be worked out giving upper bounds for the number of terms
required to secure a summation result correct to a given number of decimal
places.
APPROXIMATING SUMMATIONS WITH GIVEN ACCURACY
Assuming that we have related the summations of a given degree to one
summation, it is only necessary to consider summations of the forms
1969]
SUMMATION OF INFINITE FIBONACCI SERIES
00
157
00
n=l
^
q
n=l
q
Two cases will be taken up according as q is even or odd.
q even
From previous discussion,
< *j5/rn
1/F
l/F
n
if n is odd
< *s/5/r ~
if n is even.
For
q
the result depends on the power of r found on the right-hand side of the inequality. These powers can be calculated by table as follows.
n odd
n even
2n
n - 1
2(n + 2)
2(n + 4)
2(n + i)
2(n + 3)
2(n + q - 2 )
2(n + q - 3 )
n+q- 1
The sum in either case is
If we want w terms of the summation
158
SUMMATION OF INFINITE FIBONACCI SERIES
E
[Apr.
(-l) 1 1 - 1
n=l
^
q
to give a result correct to t decimal places, the (w + 1)
st
term must be less
than 5 x 10" . Hence the condition for the desired upper bound is:
W~T1
q
^q(w+l)+q(q-2)/2
< _
5 .
<
5 x
KT*
which leads to
t + ^ J ) l o g 5 „ a . log
_ _ — _ . — .
_
q log r
r
<
w
or
dB)
w > M | M
t +
i H ^ )
(3.34467)-f
For example, if q is 8 and we want the result correct to 10 decimal places,
w > .59814t- 2.74575 or w > 3.3565 .
Hence four terms would be required. Data from this formula will be found in
Table V.
For the summation
00
y
j ^
i
^n+q-l'q
to have the result correct to t decimal places,
1969]
SUMMATION OF INFINITE FIBONACCI SERIES
00
00
V
i
/ '
^\.
<V
£-"* '^k+a-l*
K
k=w+l
^ q
159
^
k=w+l
qk-Hi(q-2)/2
,q/2
[ l + r"q + r" 2q
rqw+q2/2
< 5 x 10 _ t
or
<5xl0_t
7 ^ 7 2 ~^
This leads to the inequality
t+iS^log5-log(l-r-q)-3iifL
w
>—
—
———
— — —
1
q log r
The term
2q
-3q
-log (1 - r""q) = r " q + - ~ - + ~
<r
.-q + r - 2 q
w>
4.78514t
=
_ L
rq-l
This replacement is in the safe direction*
(16)
+ r-3q...
+
Hence
3.34467(q - 2) _ 3
q
2q
2
+
(rq-l)qlogr
Similar considerations applied to the case of q odd give the following results.
For the summation with alternating terms:
M\
(1
"
w s 4.78514t , 3.34467 (q - 2)
>
q~" +
2q
W
(q2 - 1)
2q~
160
SUMMATION OF INFINITE FIBONACCI SERIES
[Apr.
F o r the summation with all terms positive:
n*\
w s 4.78514t
q
(18)
4.78514
(q2 - 1)
2q
q ( r q _ 1} ~
3.34467(q - 2)
2q
'
Table V
UPPER BOUNDS FOR THE NUMBER OF TERMS REQUIRED FOR RESULTS
TO t DECIMAL PLACES FOR THE SUMMATION
00
E
n=l
(-Dn
-1
T
(F
v
n+q-
q
q
t
2
4
6
8
5
11
5
3
1
10
23
11
7
4
2
15
35
17
11
7
4
2
20
47
23
15
10
6
4
2
25
59
29
19
13
9
6
3
1
30
71
35
23
16
11
8
5
3
1
50
119
59
38
28
21
16
12
9
6
4
78
58
45
36
29
24
20
16
100
239
119
10
12
14
16
18
20
T a b l e VI
U P P E R BOUNDS FOR THE NUMBER O F TERMS REQUIRED FOR RESULTS
TO t DECIMAL P L A C E S FOR THE SUMMATION
00
'j
1*> 7v p
h=l
' '
5
n+q-l'
t
2
4
6
8
10
5
13
6
3
1
10
25
12
7
4
2
12
14
16
18
20
15
37
17
11
7
4
2
20
49
23
15
10
6
4
2
25
61
29
19
13
9
6
3
1
30
73
35
23
16
11
8
5
3
1
50
121
59
39
28
21
16
12
9
6
4
100
240
119
78
58
45
36
29
24
20
16
1969]
SUMMATION OF INFINITE FIBONACCI SERIES
161
Table VH
UPPER BOUNDS FOR THE NUMBER OF TERMS REQUIRED FOR RESULTS
TO t DECIMAL PLACES FOR THE SUMMATION
00
n-1
z
n=l
q
(-D
7F~
n+q-
q
t
1
3
5
7
9
n
5
23
8
4
10
47
16
9
2
5
3
i
15
71
24
13
9
5
3
1
20
95
32
18
12
8
5
3
1
25
119
40
23
15
11
7
5
2
1
30
143
48
28
19
13
9
6
4
2
50
239
79
47
18
14
10
8
5
159
32
67
24
95
51
40
32
26
22
18
100
478
13
15
17
19
Table VIE
UPPER BOUNDS FOR THE NUMBER OF TERMS REQUIRED FOR RESULTS
TO t DECIMAL PLACES FOR THE SUMMATION
oo
WW
n=l
1
n+q-
q
9
n
5
3
I
14
9
5
3
1
32
18
12
8
5
3
1
127
40
23
15
11
7
5
2
1
48
28
19
13
9
6
4
50
151
246
80
47
32
24
18
10
5
100
486
160
95
67
51
40
14
32
2
8
26
22
18
t
1
3
5
7
5
31
8
4
2
10
55
16
9
15
79
24
20
103
25
30
13
15
17
19
These tables indicate impressively the gain in efficiency obtained by expressing
lower-degree summations in terms of ahigher degree summation. The method
of achieving this will now be taken up.
162
SUMMATION OF INFINITE FIBONACCI SERIES
[Apr.
LOWER DEGREE SUMMATIONS
IN TERMS OF HIGHER DEGREE SUMMATIONS
The program to be carried out illustrating this process will consist in
starting with
00
n=l
and establishing a chain of formulas reaching to
00
5^.
The first step in this chain is found in a result given in the Fibonacci Quarterly
[2] 9 namely:
oo
XNt-'-i;^
n~l
n=l
n=l
6
The next step is as follows.
00
^
n=l
n+2' 33
00
4—f
]_
n=
_
n
iX
00
n+1 AA
n+4
'^
^—•
~ i
"*-*•
1 • 1• 3
+
V
A/ ,
~
n=l
J
n
F F A- F , Q
n n+1 n+3
= - 1 / 3 + 1/4 = - 1 / 1 2 .
It is possible to express
M£n-1
F__ ,., F._ , n F.
n+1 n+2 n+4
1969]
SUMMATION OF INFINITE FIBONACCI SERIES
163
n-1
(-1)
F F
F
n=l n n+1 n+4
in terms of
00
b
n=l
Starting with
n-1
•(-I) 1
i F
F
L n
n=l
F
F
n+1 n+3
F
1-1-3
F
* n+1 n+2 n+4.
and noting that
F
n+2 F n+4
1/3
+ F
n F n+3 = ^
^
+ 2F
n+2
+
F
^
n=l
Hence
E
(-i)11-1
= 6i
- F n F n+l F n+4 ~
n=l
ir»
i
" 2 L4K+
n+4'
n=i
Substituting into (19),
(20)
L
n=l
(-D-n-1
n+? 3
oo
A
!L\y
n=l
n+3
n-1
n F n+1 Fn+4
( - ! ) •
(F
Zy
n44 ) 5&
n=l
F
i
5
164
SUMMATION OF INFINITE FIBONACCI SERIES
[Apr.
The next step is as follows.
^ L ( F n + 3 > 4 F n + 5 " <Fn+4>!Fn-«3 J =
I'l^-S-B
Z
r.Fm4gn46-gngi»rt1
"
00
00 _
1
= 2 >
jg
r-y
jr— + 3 2
*—4
n+2'o3 x n+5
n+5 x'n+6
n+6
A — ' F n F n+1 F n+3 F n+5 F n+6
*—f Vv t n+2'
*-*
u
n—I
n=l
,(-Dn
oy
Q+6
W
»
We shall not derive the relations for the fifth-degree summations in terms of
00
7F—r~=
n=l
(
n+4
s
*
5
but simply state them.
00
3S
J J F n+5
.
2~S
n+
r± , F n+6
• - IF
.(Fn+2«
n=l
°
1
20 " 4800
00
—^
n=i
F
n
F
1
_ 7S
F
j.i
«
^
Q
F
_
F
_
40
n+1 n+3 n+5 n+6
Substitution leads to the result
29
9600
1969]
SUMMATION O F INFINITE FIBONACCI SERIES
(21)
V*
1
- Jl_
"5=1 'n+4'5
+
165
40 X ^ (-if'1
^Wn+6V
™
The final stage i s as follows.
00
§
(_1)I1
" IKJS Fn+7 + 'We FnJ = T ^
5-8-21 '
The numerator of the combined terms inside the brackets is
F
n+6
F
n+8 + F nFn+7 =
4 F
n+4 Fn+6 + 5 Fn+4 Fn+5 +
Letting
oo
-E
~1
.,
(F
n+3*4 F n + 5 F n + 7
00
B- V .
F
n+8
-
^
and
c
=E IF^7
n=l
a
the result can be written as
M-2-3-5-8.21
Similarly,
= 4A + 5B + 3C
3
^
,n-l
166
SUMMATION OF INFINITE FIBONACCI SERIES
2 ("i)n_1
+—f
n+3V n+7 3
w~jwr^r
+
n+8Q)
r n+4
j <F_
- 1*1.2.3.8.13.21v
/ v
4
7
The numerator of the combined terms within the brackets is
n
1
- F n+4
^ F n+6
_ + 6 F n+4
^ F n+5
J_t, + 3(-l) *"
F n+4
JA F n+8
AQ + F F _
n n+5 =
leading to:
1
= -A + 6B + 3C
M-2-3.8.13.21
Solving with the previous relation in A, B, and C gives:
A
53
A = 1900080
3C
29 '
It can also be shown that
A = ± \ ^ (-1>n~1 +
73
91 Z»J ^ 7 ? ~7
2981160 °
n=l
This enables us to arrive at the final conclusion
(99.
(22)
(-I)*"1 _
X^
589
273 V ^
1
^ ^ 3 7 " ™rc*" ~^Z«f K^
n=l
'
a
n=l
Thus a chain has been extended from
00
E^
=-
n=l
to
ii
[Apr.
00
STF^n+8'
=r
n=l
•,—'a
9
1969]
SUMMATION OF INFINITE FIBONACCI SERIES
167
Connecting the initial and terminal links
L
(23)
_i_
F
„
=
46816051
13933920
16380
319
n=l
oo
n=l
Another advantage of a summation such as
00
E
l
n=l
is that it lends itself readily to calculation. At each stage one factor i s added
to one end of the denominator and deleted from the other.
Table EK shows the
calculation of this summation to thirty plus decimal places. The factor applied
at each stage to the result on the preceding line is shown at the left.
Table IX
CALCULATION OF V T ^ - T
n=l
Term
1
Factor
2
1/55
3
1/89
4
5
T e r m Multiplied by 10
xx
n+8'
33
4488
97507
72103
71328
01838
684
81
61772
86765
52205
96397
067
91705
31311
97215
79734
799
2/144
1273
68490
44405
77496
317
3/233
16
39937
64520
24602
957
21749
83614
32687
042
285
24375
26986
060
3
75700
99139
634
6
5/377
7
8/610
13/987
8
9
21/1597
4940
33864
704
10
34/2584
65
00445
588
11
55/4181
85511
721
12
89/6765
1124
988
13
144/10946
14
800
14
233/17711
SUM
59844
195
456
4571
52276
20648
18372
168
SUMMATION OF INFINITE FIBONACCI SERIES
April 1969
With the aid of this value
00
n=l
i s found to be to twenty-five decimal places
3.35988 56662 43177 55317 20113
.
CONCLUSION
In this paper two types of infinite Fibonacci series have been considered.
Methods have been developed for expressing series of the same degree in terms
of one series of that degree. In addition a path has been indicated for proceeding from series of lower degree to those of higher degree so that more rapid
convergence may be attained. These two approaches plus the development of
closed formulas in a previous article should provide an open door for additional research and calculation along the lines of sums of reciprocals of Fibonacci series of various types.
REFERENCES
1. Brother Alfred Brousseau, "Fibonacci-Lucas Infinite Series Research, M
the Fibonacci Quarterly, Vol. 7, No. 2, pp. 211-217.
2. The Fibonacci Quarterly, Problem H-10, April 1963, p. 53; Solution to
H-10, the Fibonacci Quarterly, Dec. 1963, p. 49.
• • • • •