Some Fibonacchi!!
Fibonacchi series is defined by,
i) $F_0 = F_1 = 1$
ii) $F_{i+2} = F_{i+1}+F_i$ for all $i\geq 0 $
Now, Prove that,
\[1. \sum_{i\geq 0}{\frac{F_i}{F_{i+1}F_{i+2}}} = 1\]
\[2. \sum_{i\geq 0}{\frac{1}{F_i}}\text{ converges [ from Nayel vai ]}\]
\[3. \sum_{i\geq 0}{\frac{(i+1)F_i}{F_{i+1}F_{i+2}}}\text{ also converges}\]
\[4. \sum_{i\geq 0}{\frac{F_{i+1}}{F_{i}F_{i+2}}} = 2\]
\[5. \sum_{i\geq 0}{\frac{(i+1)F_{i+1}}{F_{i}F_{i+2}}}\text{ converges too.}\]
i) $F_0 = F_1 = 1$
ii) $F_{i+2} = F_{i+1}+F_i$ for all $i\geq 0 $
Now, Prove that,
\[1. \sum_{i\geq 0}{\frac{F_i}{F_{i+1}F_{i+2}}} = 1\]
\[2. \sum_{i\geq 0}{\frac{1}{F_i}}\text{ converges [ from Nayel vai ]}\]
\[3. \sum_{i\geq 0}{\frac{(i+1)F_i}{F_{i+1}F_{i+2}}}\text{ also converges}\]
\[4. \sum_{i\geq 0}{\frac{F_{i+1}}{F_{i}F_{i+2}}} = 2\]
\[5. \sum_{i\geq 0}{\frac{(i+1)F_{i+1}}{F_{i}F_{i+2}}}\text{ converges too.}\]
ধনঞ্জয় বিশ্বাস
Re: Some Fibonacchi!!
What is Fibonacchi?
Re: Some Fibonacchi!!
Corei13 has given the definition of fibonacci series. I am just explaining it. Let the series is \[F_0,F_1,F_2,F_3...\]TOWFIQUL wrote:What is Fibonacchi?
According to the definition , $F_0=F_1=1$ and $F_{i+2}=F_i+F_{i+1}$ for any $i$.
So, \[F_2=F_0+F_1=2\]\[F_3=F_1+F_2=3\]\[...\]
The series looks like this: \[1,1,2,3,5,8,13,21,...\]
Every logical solution to a problem has its own beauty.
(Important: Please make sure that you have read about the Rules, Posting Permissions and Forum Language)
(Important: Please make sure that you have read about the Rules, Posting Permissions and Forum Language)
Re: Some Fibonacchi!!
First solution to $2.$ $\sum_{i\ge 0}\frac 1 {F_i}=4-\phi $ where $\phi $ is the golden ratio and $i$ is up to infinity
Second solution:Let $u_r=\frac 1 {F_r}$ and it is enough to prove that $\frac {u_{r+1}} {u_r}<1$ for $r$ tends to infinity and the limit be $l$.Then $l=\frac 1 {\phi }<1$,so it converges
Second solution:Let $u_r=\frac 1 {F_r}$ and it is enough to prove that $\frac {u_{r+1}} {u_r}<1$ for $r$ tends to infinity and the limit be $l$.Then $l=\frac 1 {\phi }<1$,so it converges
Last edited by Masum on Tue Jan 04, 2011 7:52 pm, edited 1 time in total.
Reason: Edit:Since a confusion arose due to this theorem(which I described in the book of Arthur Engel),I am editing this without the word well-known.
Reason: Edit:Since a confusion arose due to this theorem(which I described in the book of Arthur Engel),I am editing this without the word well-known.
One one thing is neutral in the universe, that is $0$.
Re: Some Fibonacchi!!
Can you give the proof/method for $\sum_{i\geq 0}{\frac{1}{F_i}} = 4 - \phi$ ?
ধনঞ্জয় বিশ্বাস
Re: Some Fibonacchi!!
Hint for $1:$ Use $F_i=F_{i+2}-F_{i+1}$ to telescope the series
One one thing is neutral in the universe, that is $0$.
Re: Some Fibonacchi!!
Can you give a proof/link to a proof, if this is so well-known?Masum wrote:First solution to $2.$ It is well-known that $\sum_{i\ge 0}\frac 1 {F_i}=4-\phi $ where $\phi $ is the golden ratio and $i$ is up to infinity
"Everything should be made as simple as possible, but not simpler." - Albert Einstein
Re: Some Fibonacchi!!
I am surprised...why don't people give complete solutions to these cool problems?
Just use what Galileo invented...the telescope
Just use what Galileo invented...the telescope
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Re: Some Fibonacchi!!
This theorem is cited in the book $\text {Problem Solving Strategies}$,so we can take it for a well-known one of-course(Chapter 8,Induction Principle)
One one thing is neutral in the universe, that is $0$.
Re: Some Fibonacchi!!
According to wikipedia, no closed formula for the sum of the reciprocals of the Fibonacci numbers is known. So if you can prove that, your name will surely be remembered for a long time in the history of mathematics!
"Everything should be made as simple as possible, but not simpler." - Albert Einstein