Some Fibonacchi!!

[Corei13]

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,
\[1. \sum_{i\geq 0}{\frac{F_i}{F_{i+1}F_{i+2}}} = 1\]
\[2. \sum_{i\geq 0}{\frac{1}{F_i}}\text{ converges [ from Nayel vai ]}\]
\[3. \sum_{i\geq 0}{\frac{(i+1)F_i}{F_{i+1}F_{i+2}}}\text{ also converges}\]
\[4. \sum_{i\geq 0}{\frac{F_{i+1}}{F_{i}F_{i+2}}} = 2\]
\[5. \sum_{i\geq 0}{\frac{(i+1)F_{i+1}}{F_{i}F_{i+2}}}\text{ converges too.}\]

[TOWFIQUL]

What is Fibonacchi?

[Zzzz]

What is Fibonacchi?

Corei13 has given the definition of fibonacci series. I am just explaining it. Let the series is \[F_0,F_1,F_2,F_3...\]
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: \[1,1,2,3,5,8,13,21,...\]

[Masum]

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

[Corei13]

Can you give the proof/method for $\sum_{i\geq 0}{\frac{1}{F_i}} = 4 - \phi$ ?

[Masum]

Hint for $1:$ Use $F_i=F_{i+2}-F_{i+1}$ to telescope the series

[nayel]

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

Can you give a proof/link to a proof, if this is so well-known?

[Moon]

I am surprised...why don't people give complete solutions to these cool problems?
Just use what Galileo invented...the telescope

[Masum]

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)

[nayel]

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!