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okay in this video I want to consider the Fibonacci numbers so theyre defined by this very well-known recursion so we have f of 0 equals 0 F of 1 equals 1 and then we have this two-step recursion F of n equals F sub n minus 1 plus F sub n minus 2 for all n bigger than or equal to 2 and so our main goal for the video is to find this sum N equals 0 to infinity of F sub n over 2 to the N and were going to do this two different ways were going to do this first with a generating function which I think is maybe the standard way as a bonus to that were going to find the sum of n goes from 0 to infinity of n times F sub n over 2 to the N and then were going to use this double sum technique which I just learned in a very recent video by black and red pen so Im pretty excited about to check that out okay so now lets go ahead and look at this generating function approach so what we want to do is set this function capital f of X equal to the sum N equals 0 to infinity of F sub n X to the N