Simply take last 2 numbers in sequence and sum them
0 + 1 = 1
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
8 + 13 = 21
13 + 21 = 34
Etc
Now, as you go up the ladder of ratios
Of course 1/2 < 2/1 - 1
2/3 > 3/2 -1
3/5 < 5/3 - 1
5/8 > 8/5 - 1
8/13 < 13/8 - 1
As this sequence goes on indefinitely
You approach
X/Y = Y/X - 1 = The Golden Ratio
If you take reciprocals
1/1 + 1/1 + 1/2 + 1/3 + 1/5 + 1/8 + 1/13 + 1/21.....
Converges to a unique constant greater than pi and less than 4, but more than
1 / (1 - golden ratio) Geometric series 1/(1-R)
Obviously because the early terms have not converged on this ratio,
But
What constant do we get if we take only the reciprocals of the Odd Fibonacci numbers?
Exclude 2, 8, 34, 144....
2/3 of the Fibonacci numbers are odds actually because odd + odd = even . Odd + even = odd.
Even + odd = odd
Odd + odd = even
2 odds to every even
What is the convergence infinite series only odd Fibonaccis? Is it less than e? Is it less than 1/ 1-(golden ratio)
0 + 1 = 1
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
8 + 13 = 21
13 + 21 = 34
Etc
Now, as you go up the ladder of ratios
Of course 1/2 < 2/1 - 1
2/3 > 3/2 -1
3/5 < 5/3 - 1
5/8 > 8/5 - 1
8/13 < 13/8 - 1
As this sequence goes on indefinitely
You approach
X/Y = Y/X - 1 = The Golden Ratio
If you take reciprocals
1/1 + 1/1 + 1/2 + 1/3 + 1/5 + 1/8 + 1/13 + 1/21.....
Converges to a unique constant greater than pi and less than 4, but more than
1 / (1 - golden ratio) Geometric series 1/(1-R)
Obviously because the early terms have not converged on this ratio,
But
What constant do we get if we take only the reciprocals of the Odd Fibonacci numbers?
Exclude 2, 8, 34, 144....
2/3 of the Fibonacci numbers are odds actually because odd + odd = even . Odd + even = odd.
Even + odd = odd
Odd + odd = even
2 odds to every even
What is the convergence infinite series only odd Fibonaccis? Is it less than e? Is it less than 1/ 1-(golden ratio)
Question on Fibonacci Numbers 1 2 3 5 8 13 21 34 55 89...