Reading on with Julian Havil’s Gamma: Exploring Euler’s Constant and inspired by his discussion of the harmonic series, I come across this:
Havil calls this a “non-legitimate binomial expansion” and it seems to me it can be generalised:
And, indeed if we take we get:
at the limit.
But if we have it is divergent and the identity, which seems algebraically sound to me, breaks down. E.g., :
So what is the flaw in my logic?