[seqfan] Asymptotic behaviour of a sum

[Bernard Cloitre]

Dear seqfans,

Looking for properties of a rational sequence I came across the following
finite sum:

S(n)=sum_{0<=k<=ceil(log(n)/log(2))} (-1)^k*log(n/2^k)^(k+1)/(k+1)!

Then S(n)-->1 as n-->infty and I suspect there is a real value alpha>0 such
that:

 1-S(n)=O(n^(-alpha)).

Experiments suggest alpha exists and is closed to 0.82.... Does somebody see
what could be the exact value of alpha (theorically or empirically)?

Thanks

Benoit