The last three terms of 
in equation 4.12 for 
with B/N fixed are 
. Specifically, expanding these correction terms in a Taylor series for N with B=N q yields
Note 1) this is independent of q only in the first term; 2) this is negative, which is required to satisfy the theorem of section 4.7;
3) in deriving this result it was assumed that 
, so that the second order entropy exists; however 4) in deriving this result a range for B was
not assumed other than 
, contrary to the perhaps misleading form of the correction terms in equation 4.12; 5) the correction terms of equation 4.12 are not necesarily real for noninteger N and
B or, therefore, away from the discrete values of q allowed. However it is clear from the form of equation 4.12 that the correction terms become imaginary for noninteger B only for 0<B<1 (0<q<1/N) and 0<N-B<1
(1-1/N<q<1); 6) the third and higher order correction terms are not independent of q; 7) the third order correction term becomes singular at q=0,1 (B=0,N), indicating that for fixed N, as 
(
), the correction terms can become quite large. However, entropies are always positive, and as all entropies converge to zero, so either the series for the correction terms must also converge to
zero in sum in these limits, or the radius of convergence of the series does not include these limits, which is actually the case; 8) each coefficient of
the series of equation 4.16 is a real function of q, yet the overall correction of equation 4.12 is not real for values of q in the ranges mentioned above. This is an effect of the fact that the radius of
convergence of the Taylor series includes only those values of N for which the arguments of the logarithms in equation 4.12 are all positive.
Not unexpectedly after the discussion just given, the maximum absolute value of the correction of equation 4.12 occurs at the points q=1/N,1-1/N, and has a slope which diverges logarithmically as these points are approached. In this unphysical sense, the derivative of the second order entropy with respect to bit density shows a phase transition as the number of one bits per string of length N drops to one, or increases to N-1. Again, what this indicates physically is not defined, as it is not possible to have fractional bits.