Exercise 5, Appendix 0

a) Deduce from exerc. 4 that if $f$ is an (H) function having a definition sequence of $n$ terms, not equivalent to a constant, and of order 0 relative to $l_{n-1}(x)$, there exist a constant $k$ and a rational integer $r$ such that $f(x) \sim k(l_n(x))^r$.

b) Deduce from exerc. 4 that if $f$ is any (H) function there exists an integer $n$ such that the logarithmic criterion of order $n$ will determine whether the integral $\int_a^{+\infty} f(t)\,dt$ converges or is infinite.