  Let $\gamma_n$ ($n\in \mathbb{Z}_{\ge0}$) be a sequence of complex numbers,
which is tame: $0<\exists u\le \gamma_{n-1}/\gamma_n \le \exists v<\infty$ for
all $n>0$. We show a resonance between the singularities of the function of the
power series $P(t):=\sum_{n=0}^\infty \gamma_n t^n$ on its boundary of the disc
of convergence and the oscillation behavior of the sequences
$\gamma_{n-k}/\gamma_n$ ($n\in \mathbb{Z}_{>>0}$) for $k>0$. The resonance is
proven by introducing the space of opposite power series, which is the compact
subspace of the space of all formal power series in the opposite variable
$s=1/t$ and is defined as the accumulating set of the sequence
$X_n(s):=\sum_{k=0}^n\frac{\gamma_{n-k}}{\gamma_n}t^k$ ($n\in
\mathbb{Z}_{\ge0}$). We analyze in details an example of the growth series
$P(t)$ for the modular group $PSL(2,Z)$ due to Machi.
