Fwd: . Graphical Proof of convergence of the Ratio of consecutive Fibonacci Numbers . 09 apr 2017 sun 0727 am PDT .

[h.s.n...@gmail.com]

On Monday, April 10, 2017 at 9:43:30 AM UTC-7, h.s.n...@gmail.com wrote:

. date 10 apr 2017 mon 0908 am PDT .
. topic . Graphical Proof of convergence of the Ratio of consecutive Fibonacci Numbers . 09 apr 2017 sun 0727 am PDT .
. Proof . Fibonacci Numbers may be represented by the following sequence . 1 1 … a b a+b … where the sequence of numbers starts with 1 1 and the next number in the infinite sequence is given by the sum of the last two numbers . The first few numbers in the sequence may be enumerated as follows . 1 1 2 3 5 8 13 21 33 54 … The reverse ratio of consecutive Fibonacci numbers may be defined by x = Fn / Fn+1 = a / b and then Fn+1 / Fn+2 = b / ( a + b ) = 1 / ( 1 + ( a  /  b ) = 1 / ( 1 + x ) where Fn Fn+1 Fn+2 are consecutive Fibonacci Numbers . This means the reverse ratio of consecutive Fibonacci Numbers follows a sequence defined as follows . If Rn = x then Rn+1 = 1 / ( 1 + x ) with R1 = 1 . In order to prove that the sequence of reverse ratios Rn with n = 1 to infinity starting with R1 = 1 will converge we may model the sequence of reverse ratios as an iteration between 2 curves y1 ( x ) = x and y2 ( x ) = 1 / ( 1 + x ) starting with y1 = 1 as point 1 and iterating forward to y2 = 1 / ( 1 + 1 ) = 1 / 2 = 0.5 as point 2 and then y1 = 0.5 as point 3 and then y2 = 1 / ( 1 + 0.5 ) = 0.666 … as point 4 and then y1 = 0.666 … as point 5 and then y2 = 1 / ( 1 + 0.666 … ) = 0.6 as point 6 and so on until the sequence of reverse ratios Rn will converge in a shortening spiral at the intersection of the 2 curves at the point where x = 1 / ( 1 + x ) = 0.61803398875 . end of proof . A one pager showing the graphical representation of the two curves and the convergence of the sequence of ratios is shown at the following link . https://drive.google.com/file/d/0B7lF1hUXdsV3dkVjMXlmMUtXbVE/view?usp=sharing . This is a geometric or graphical proof of convergence that can be used for any two monotonic curves that intersect . I claim that this is a new method for proving convergence of sequences . I heard on the air on 08 apr 2017 sat that Princeton University , Princeton , New Jersey , USA , has awarded me a Doctorate in Mathematics for my post in the internet newsgroup soc.culture.indian of 07 apr 2017 fri copied below for reference entitled . What if the ratio of consecutive Fibonacci Numbers did converge . Please send me your comments if any . My email is h.s.n...@gmail.com . Thank you . Hemarajan Shankaranarayanan Nair .

. References . my previous post of 07 apr 2017 fri in this internet newsgroup soc.culture.indian is shown below .

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Subject: . What if the ratio of consecutive Fibonacci Numbers did converge .

 07 apr 2017 fri 0931 am PDT .

From: h.s.n...@gmail.com

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. What if the ratio of consecutive Fibonacci Numbers did converge . 07 apr 2017 fri 0931 am PDT .
. date 07 apr 2017 fri 0931 am PDT .
. topic . What if the ratio of consecutive Fibonacci Numbers did converge .
. Fibonacci numbers are defined by the sequence … a b a+b … starting with 1 1 . Consecutive Fibonacci numbers are obtained by adding the last two numbers to get the next number . The first few of them may be enumerated as 1 1 2 3 5 9 13 21 33 54 … I read a post in this internet newsgroup soc.culture.indian dated 03 apr 2017 by DrJM titled . Using Fibonacci Numbers to convert from miles to kilometers and vice versa . and discussed at the website

http://www.catonmat.net/blog/using-fibonacci-numbers-to-convert-from-miles-to-kilometers/

. The discussion is about the ratio of consecutive Fibonacci numbers converging to a number 1.6 that is also the ratio relating 1 mile to 1 kilometer . Upon further investigation I find that it is possible to mathematically arrive at the exact ratio . If the forward ratio is defined as Fn+1 divided by Fn and the reverse ratio is defined as Fn divided by Fn+1 where Fn and Fn+1 are consecutive Fibonacci numbers then we can see that for the sequence a b a+b the forward ratio would be  ( a + b ) / b or 1 + ( a / b ) or 1 + x where x is the fraction a / b . The reverse ratio for these numbers would be a / b or x . Now we may state that if the two ratios converge then as the numbers tend to infinity the reciprocal of the forward ratio would approach the reverse ratio . This means 1 / ( 1 + x ) = x for large numbers in the sequence .This is a quadratic equation ( x squared ) + x - 1 = 0  with roots x = ( -1 ( + or - ) sqrt ( 5 ) ) / 2 . That gives x = 0.61803398875 since the negative root is not meaningful for the sequence . You may verify that ( 1 / x ) = 1.61803398875 = 1 + x as I found using the google.com online calculator . That means if the ratio of consecutive Fibonacci numbers converges then it would converge to a forward ratio of 1.61803398875 and a reverse ratio of 0.61803398875 . end of message . Please send me your comments if any . My email is h.s.n...@gmail.com . Thank you . Hemarajan Shankaranarayanan Nair .