Why is this proposition a corollary of this theorem?

Question

I am reading "A First Course in Analysis vol.1" by Sin Hitotumatu.

There is a theorem in this book:

Theorem 7.10
For any power series $a_0 + a_1 x + a_2 x^2 + \cdots a_n x^n + \cdots$, there exists the unique non-negative real number $\rho$ such that
$a_0 + a_1 x + a_2 x^2 + \cdots a_n x^n + \cdots$ absolutely converges for any $x$ such that $|x| < \rho$ and $a_0 + a_1 x + a_2 x^2 + \cdots a_n x^n + \cdots$ diverges for any $x$ such that $|x| > \rho$.

And this is a corollary of the above theorem:

Corollary
A power series $a_0 + a_1 x + a_2 x^2 + \cdots a_n x^n + \cdots$ has a positive radius of convergence
$\Leftrightarrow$
$|a_n| < c M^n$ for all $n$ for some positive real numbers $c$ and $M$.

Why is this proposition a corollary of the theorem 7.10?

Answer 1

For $\implies$ note that $\sum a_n (\frac p 2)^{n}$ converges. Hence the general term $a_n (\frac p 2)^{n}$ tends to $0$. So there exists $c$ such that $|a_n (\frac p 2)^{n}| \leq c$. This gives $|a_n| \leq (\frac 2 p )^{n}c$.

The converse part is easy and I will let you handle that.

Answer 2

Let $P(x) = a_0 + a_1 x + \ldots$ and choose an $\alpha$ such that $P(\alpha)$ absolutely converges. Then for any given real we must have $a_n \alpha^n$ eventually being below this real after a certain point. So for some $N$ we have $|a_n \alpha^n| < 1$ for all $n>N$, which gives $|a_n| < M^n$ where $M = \frac{1}{\alpha}$. Choosing a sufficiently large $c$ will deal with all terms $n \leq N$.