Jonathan, this is an interesting task, but you say:
The direct attempt to find the limit of sum of series doesn’t succeed, ...
If I try that on my system
$Version
(* Out: "12.3.1 for Linux x86 (64-bit) (June 24, 2021)" *)
I get:
rezipSum = Sum[1/Fibonacci[n], {n, 1, Infinity}]
N[rezipSum, 10]
(* Out: 3.359885666 *)
This is a nice closed solution which already contains the GoldenRatio. Isn't Mathematica just awesome?!?
Which version are you using? Obviously WRI made some improvements!
