Sunday, November 15, 2015

The Series ( ((((0 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + … + 1)^-1 and Fibonacci Numbers

The Series ( ((((0 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + … + 1)^-1 and Fibonacci Numbers
  


Add One Then Reciprocate

Define the series t as:

t = ( ((((0 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + … + 1)^-1    (an infinite amount of terms)

This is a sum that can’t be easily stated in summation statement (Σ f(x)). 

On the HP Prime, I programmed this as:

EXPORT TEST1112(n)
BEGIN
LOCAL k, t:=0;
FOR k FROM 1 TO n DO
t:=(t+1)^-1;
END;
RETURN t;
END;

The result seems to converge at 0.6180339785 when n ≥ 27.  Note that 0.6180339785 = ϕ – 1, where ϕ is the Golden Ratio ( ϕ = (√5 + 1)/2)






Fibonacci Gets Involved

Note that:

k =
t =
1
1
2
(1 + 1)^-1 = 1/2
3
(1 + 1/2)^-1 = (3/2)^-1 = 2/3
4
(1 + 2/3)^-1 = (5/3)^-1 = 3/5
5
(1 + 3/5)^-1 = (8/5)^-1 = 5/8
6
(1 + 5/8)^-1 = (13/8)^-1 = 8/13
7
(1 + 8/13)^-1 = (21/13)^-1 = 13/21

We get a sequence of terms {1, 1/2, 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55, 55/89, 89/144, …} where each term takes the fraction a/b, a is the kth Fibonacci number and b is the (k+1)th Fibonacci number.  Can we show that this sequence of partial sums is convergent?

Each partial sums of the series takes the form F_k / F_k+1 where F is the Fibonacci number.

The closed formula for the Fibonacci number is:

F_k = ( ϕ^k – α^k )/√5 , where ϕ  = (1 + √5)/2 and α = (1 - √5)/2.

Then:

F_k / F_k+1
=( ϕ^k – α^k )/√5 * √5/( ϕ^k+1 – α^k+1)
=( ϕ^k – α^k) / ( ϕ^k+1 – α^k+1 )
= ( ϕ^k / ϕ^k+1) * ( (1 – (α/ϕ)^k) / (1 – (α/ϕ)^k+1) )
= 1/ϕ * ( (1 – (α/ϕ)^k) / (1 – (α/ϕ)^k+1) )

Note that α/ϕ = (1 - √5)/(1 + √5 ) ≈ -0.38197 < 1

As k → ∞,  α/ϕ → 0. 

Hence,

lim k → ∞ (F_k / F_k+1)
= lim k → ∞ (1/ϕ * ( (1 – (α/ϕ)^k) / (1 – (α/ϕ)^k+1) ) )
= 1/ϕ

Simplifying:

1/ϕ
= 2/(1 + √5)
= 2*(1 - √5) / ((1 + √5)*(1 - √5))
= 2*(1 - √5)/-4
= (√5 – 1)/2
= √5/2 – 1/2

Adding and subtracting 1/2:

√5/2 – 1/2 + (1/2 – 1/2)
= (√5 + 1)/2 – 1
= ϕ – 1

Since the sequence of partial sums converge to ϕ – 1, the series

t = ( ((((0 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + 1)^-1 + … + 1)^-1   

converges to ϕ – 1.



This blog is property of Edward Shore.  2015.



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