Browsing this web page I stike your question and since it is an oldie one I do not know if you got the answer yet.
In case you are not it that it is the approach that will give the sum as close as you want.
1) You probably should know that the ratio between to consecutive Fibonacci number F(k+1)/F(k) become phi (the Golden Ratio number0 as k approach to infinite. This ratio is oscilating what it is a good news so if F(k+1)/F(k)> phi means that F(k+2)/F(k+1)< phi. Besides its converge relatively fast.
2) So all you have to do is sum a certain amount of inverse of Fibonacci number in your serie and then uses the formula of the geometric series using inverse of phi as a ratio. The result you get it is not the exact but it is in between the one you get using one term more in your series and one term less remenber as said above the ratio is oscillating. Of course you can make it as small as you want.
I leaves the computation to you since you may have already get the solution if it is not the case come back