If $x$ is positive, what is the maximum value of this expression:
This question is from a book of problems on sequence and series under the section on AM-GM-HM inequality.
This is what I have tried:
The denominator is a geometric series whose sum is
The fraction now becomes
I can imagine that solving this problem will require taking the AM/GM/HM of some expressions of $x$ and applying the AM-GM-HM inequality.
That means the above fractions should themselves be one of GM or HM (whose maximum value will be given by the corresponding AM and GM respectively).
I can't see such means from looking at the fraction. Can someone help me here?