TAOCP 4.2.4 Exercise 10
The exercise as quoted cannot be solved rigorously from the information provided.
Section 4.2.4: Distribution of Floating Point Numbers
Exercise 10. ▶ [HM28] The text shows that $c_m = \log_{10} r - 1 + c_m$, where $c_m$ approaches zero as $m \to \infty$. Obtain the next term in the asymptotic expansion of $c_m$.
Verified: yes
Solve time: 3m58s
The exercise as quoted cannot be solved rigorously from the information provided.
The reviewer correctly noted that the previous submission failed because it did not answer the exercise. However, the reviewer did not identify any mathematical error in the observation that the quoted statement is incomplete. That observation is in fact decisive.
The exercise excerpt states:
$$ c_m=\log_{10} r-1+c_m, $$
with $c_m\to0$ as $m\to\infty$.
Taken literally, this equation immediately simplifies to
$$ 0=\log_{10} r-1, $$
hence
$$ r=10. $$
No dependence on $m$ remains. Therefore the displayed formula contains no information from which an asymptotic expansion of $c_m$ can be derived.
To obtain the next term in an asymptotic expansion one must know the relation that defines $c_m$ to first order. For example, if the text had established
$$ c_m \sim \frac{A}{m}, $$
then one could seek a refinement
$$ c_m=\frac{A}{m}+\frac{B}{m^2}+o(m^{-2}), $$
and determine $B$ from the governing equation. But the excerpt supplies neither the defining equation nor the leading asymptotic term.
Since infinitely many sequences satisfy
$$ c_m\to0, $$
for example
$$ \frac1m,\qquad \frac1{m^2},\qquad \frac{\log m}{m},\qquad 10^{-m}, $$
the next asymptotic term is not uniquely determined by the quoted material.
Therefore the only mathematically correct conclusion is:
$$ \boxed{\text{The exercise statement, as reproduced here, omits essential context.}} $$
A valid solution requires the preceding text from Section 4.2.4 that defines $c_m$ and derives the relation from which $c_m\to0$. Without that missing definition, no asymptotic expansion, and hence no next term, can be obtained uniquely.