Inequality for Fibonacci to find an upper bound of harmonic Fibonacci series I want to find an sharp upper bound for $$\sum_{n=1}^{\infty}\frac{1}{F_n}$$which $F_n $ is the n$th$ term of Fibonacci sequence .
I wrote a Matlab program to find an upper bound ,$\sum_{n=1}^{10^6}\frac{1}{F_n}<4$
  Now my question is:(1):Is there an inequality to find this ?
(2): Is that series have a close form ?
$${F_n} = \frac{{{\varphi ^n} - {{( - \varphi )}^{ - n}}}}{{\sqrt 5 }}\to \\\sum_{n=1}^{\infty}\frac{1}{F_n}=\sum_{n=1}^{\infty}\frac{\sqrt 5}{{{\varphi ^n} - {{( - \varphi )}^{ - n}}}}\\\leq \sum_{n=1}^{\infty}\frac{\sqrt 5}{{{(\frac{{1 + \sqrt 5 }}{2} ) ^n} }}=\frac{\sqrt5}{1-\frac{1}{\frac{{1 + \sqrt 5 }}{2}}}\approx12.18\\$$I am thankful for a hint or solution which can bring a sharper upper bound .

Using Wolfram Mathematica answer is :
Wolfram Mathematica Code

